Math formulas cheat sheet

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Short multiplication formulas

Commutative law of addition

Grade 5
a+b=b+aa + b = b + a

Swapping the addends does not change the sum.

Legend:
  • a — Term a
  • b — Term b

Commutative law of multiplication

Grade 5
a⋅b=b⋅aa \cdot b = b \cdot a

Swapping the factors does not change the product.

Legend:
  • a — Factor a
  • b — Factor b

Distributive law

Grade 5
a⋅(b+c)=a⋅b+a⋅ca \cdot (b + c) = a \cdot b + a \cdot c

Multiplying a number by the sum of two terms.

Legend:
  • a — Number a
  • b — Number b
  • c — Number c

Associative law of addition

Grade 5
(a+b)+c=a+(b+c)(a+b)+c=a+(b+c)

The result of addition does not depend on how the addends are grouped.

Legend:
  • a — Term a
  • b — Term b
  • c — Term c

Associative law of multiplication

Grade 5
(a⋅b)⋅c=a⋅(b⋅c)(a \cdot b) \cdot c = a \cdot (b \cdot c)

The result of multiplication does not depend on how the factors are grouped.

Legend:
  • a — Factor a
  • b — Factor b
  • c — Factor c

Square of a sum

Grade 7
(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2

Short-multiplication formula for the square of a sum.

Legend:
  • a — Number a
  • b — Number b

Square of a difference

Grade 7
(a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2

Short-multiplication formula for the square of a difference.

Legend:
  • a — Number a
  • b — Number b

Difference of squares

Grade 7
a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)

Factoring the difference of two squares.

Legend:
  • a — Number a
  • b — Number b

Cube of a sum

Grade 7
(a+b)3=a3+3a2b+3ab2+b3(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

The cube of the sum of two expressions.

Legend:
  • a — Number a
  • b — Number b

Cube of a difference

Grade 7
(a−b)3=a3−3a2b+3ab2−b3(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

The cube of the difference of two expressions.

Legend:
  • a — Number a
  • b — Number b

Sum of cubes

Grade 7
a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Factoring the sum of two cubes.

Legend:
  • a — Number a
  • b — Number b

Difference of cubes

Grade 7
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Factoring the difference of two cubes.

Legend:
  • a — Number a
  • b — Number b

Equations and Parabolas

Distance formula

Grade 5
S=v⋅tS = v \cdot t

Distance travelled during uniform motion.

Legend:
  • v — Speed (v)
  • t — Time (t)

Speed formula

Grade 5
v=Stv = \frac{S}{t}

Speed as the ratio of distance to time.

Legend:
  • S — Distance (S)
  • t — Time (t)

Time of motion formula

Grade 5
t=Svt = \frac{S}{v}

Time of motion as the ratio of distance to speed.

Legend:
  • S — Distance (S)
  • v — Speed (v)

Speed downstream

Grade 5
v=v0+v1v = v_0 + v_1

Sum of the object's own speed and the current's speed.

Legend:
  • v_self — Own speed
  • v_flow — Current speed

Speed upstream

Grade 5
v=v0−v1v = v_0 - v_1

Difference between the object's own speed and the current's speed.

Legend:
  • v_self — Own speed
  • v_flow — Current speed

Cost of a purchase

Grade 5
C=a⋅nC = a \cdot n

Total cost as the price per unit multiplied by the quantity.

Legend:
  • a — Unit price (a)
  • n — Count (n)

Work (productivity)

Grade 5
A=v⋅tA = v \cdot t

Amount of work done, expressed through rate and time.

Legend:
  • v — Rate of work (v)
  • t — Time (t)

Division with remainder

Grade 5
a=b⋅q+ra = b \cdot q + r

Dividend expressed through the divisor, quotient and remainder.

Legend:
  • b — Divisor (b)
  • q — Quotient (q)
  • r — Remainder (r)

Arithmetic mean

Grade 5
M=a1+a2+⋯+annM = \frac{a_1 + a_2 + \dots + a_n}{n}

Calculates the arithmetic mean of two or three given numbers.

Legend:
  • a1 — Number 1
  • a2 — Number 2
  • a3 — Number 3 (optional: 0)

Fundamental property of a fraction

Grade 6
ab=a⋅cb⋅c\frac{a}{b} = \frac{a \cdot c}{b \cdot c}

Multiplying the numerator and denominator by c keeps the fraction's value unchanged.

Legend:
  • a — Numerator a
  • b — Denominator b
  • c — Factor c (c≠0)

Multiplying common fractions

Grade 6
ab⋅cd=a⋅cb⋅d\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

The numerators and denominators are multiplied separately.

Legend:
  • a — Numerator of the first fraction
  • b — Denominator of the first fraction
  • c — Numerator of the second fraction
  • d — Denominator of the second fraction

Dividing common fractions

Grade 6
ab:cd=a⋅db⋅c\frac{a}{b} : \frac{c}{d} = \frac{a \cdot d}{b \cdot c}

Multiplying the first fraction by the reciprocal of the second.

Legend:
  • a — Numerator of the first fraction
  • b — Denominator of the first fraction
  • c — Numerator of the second fraction
  • d — Denominator of the second fraction

Fundamental property of a proportion

Grade 6
ab=cd⇒a⋅d=b⋅c\frac{a}{b} = \frac{c}{d} \Rightarrow a \cdot d = b \cdot c

The product of the extreme terms of a proportion equals the product of the middle terms.

Legend:
  • a — Extreme term (1st)
  • b — Middle term (1st)
  • c — Middle term (2nd)
  • d — Extreme term (2nd)

Absolute value of a number

Grade 6
∣a∣={a,a≥0−a,a<0|a| = \begin{cases} a, & a \ge 0 \\ -a, & a < 0 \end{cases}

The absolute value of a number (its distance from zero on the number line).

Legend:
  • a — Number a

Adding numbers with different signs

Grade 6
a+(−b)=a−ba + (-b) = a - b

Adding a negative number is the same as subtracting its absolute value.

Legend:
  • a — Number a
  • b — Absolute value of negative b

Multiplying negative numbers

Grade 6
(−a)⋅(−b)=a⋅b(-a) \cdot (-b) = a \cdot b

The product of two negative numbers is always positive.

Legend:
  • a — Absolute value of the first number (a)
  • b — Absolute value of the second number (b)

Multiplying numbers with different signs

Grade 6
(−a)⋅b=−(a⋅b)(-a) \cdot b = -(a \cdot b)

The product of numbers with different signs is always negative.

Legend:
  • a — Absolute value of the first (a)
  • b — Absolute value of the second (b)

GCD (greatest common divisor)

Grade 6
gcd⁡(a,b)\gcd(a, b)

The largest number that divides both numbers with no remainder.

Legend:
  • a — Number a
  • b — Number b

LCM (least common multiple)

Grade 6
lcm⁡(a,b)=a⋅bgcd⁡(a,b)\operatorname{lcm}(a,b) = \frac{a \cdot b}{\gcd(a,b)}

The smallest number that is a multiple of both numbers.

Legend:
  • a — Number a
  • b — Number b

Adding fractions with the same denominator

Grade 6
ac+bc=a+bc\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}

The numerators are added, the denominator stays the same.

Legend:
  • a — Numerator a
  • b — Numerator b
  • c — Denominator c

Subtracting fractions with the same denominator

Grade 6
ac−bc=a−bc\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}

The numerators are subtracted, the denominator stays the same.

Legend:
  • a — Numerator a
  • b — Numerator b
  • c — Denominator c

Finding an unknown term of a proportion

Grade 6
ab=cx⇒x=b⋅ca\frac{a}{b}=\frac{c}{x} \Rightarrow x=\frac{b \cdot c}{a}

Finds the fourth (unknown) term of a proportion.

Legend:
  • a — First term of the proportion (a)
  • b — Second term of the proportion (b)
  • c — Third term of the proportion (c)

Map scale

Grade 6
M=DdM = \frac{D}{d}

Shows how many times the real-world distance is greater than the distance on the map.

Legend:
  • d — Distance on the map (d)
  • D — Real-world distance (D)

Linear function

Grade 7
y=kx+by = kx + b

Calculates the value of y from the argument x, slope k and offset b.

Legend:
  • k — Coefficient k
  • x — Argument x
  • b — Constant term b

Linear equation (root)

Grade 7
ax+b=0⇒x=−baax+b=0 \Rightarrow x=-\frac{b}{a}

Root of a linear equation in one variable.

Legend:
  • a — Coefficient a
  • b — Coefficient b

System of two linear equations

Grade 7
x=c1b2−c2b1a1b2−a2b1, y=a1c2−a2c1a1b2−a2b1x=\frac{c_1b_2-c_2b_1}{a_1b_2-a_2b_1},\ y=\frac{a_1c_2-a_2c_1}{a_1b_2-a_2b_1}

Solving a system of two linear equations in two variables using determinants.

Legend:
  • a1 — Coefficient a1
  • b1 — Coefficient b1
  • c1 — Coefficient c1
  • a2 — Coefficient a2
  • b2 — Coefficient b2
  • c2 — Coefficient c2

Discriminant

Grade 8
D=b2−4acD = b^2 - 4ac

Determines the number of real roots of a quadratic equation ax² + bx + c = 0.

Legend:
  • a — Coefficient a
  • b — Coefficient b
  • c — Coefficient c

Roots of a quadratic equation

Grade 8
x1,2=−b±D2ax_{1,2} = \frac{-b \pm \sqrt{D}}{2a}

Finding the roots of an equation using the discriminant.

Legend:
  • a — Coefficient a
  • b — Coefficient b
  • c — Coefficient c

Vieta's formulas (reduced form)

Grade 8
x1+x2=−p,x1⋅x2=qx_1 + x_2 = -p, \quad x_1 \cdot x_2 = q

Relates the roots of x² + px + q = 0 to its coefficients.

Legend:
  • p — Coefficient p
  • q — Coefficient q

Factoring a quadratic trinomial

Grade 8
ax2+bx+c=a(x−x1)(x−x2)ax^2 + bx + c = a(x - x_1)(x - x_2)

Expressing a trinomial as a product using its roots.

Legend:
  • a — Coefficient a
  • x1 — Root x1
  • x2 — Root x2

Vieta's formulas (general case)

Grade 8
x1+x2=−ba,x1⋅x2=cax_1+x_2=-\frac{b}{a},\quad x_1 \cdot x_2=\frac{c}{a}

Relates the roots of the quadratic ax² + bx + c = 0 to its coefficients.

Legend:
  • a — Coefficient a
  • b — Coefficient b
  • c — Coefficient c

Inverse proportionality function

Grade 8
y=kxy = \frac{k}{x}

Value of the inverse-proportionality function at a point x.

Legend:
  • k — Coefficient k
  • x — Argument x

Vertex of a parabola (x)

Grade 9
x0=−b2ax_0 = -\frac{b}{2a}

Calculates the x-coordinate of the vertex of the parabola ax² + bx + c.

Legend:
  • a — Coefficient a
  • b — Coefficient b

Vertex of a parabola (y)

Grade 9
y0=ax02+bx0+cy_0 = a x_0^2 + b x_0 + c

Calculates the y-coordinate of the vertex from x₀ and the coefficients.

Legend:
  • a — Coefficient a
  • b — Coefficient b
  • c — Coefficient c
  • x0 — Coordinate x0

Median of a sample (5 numbers)

Grade 9
Me=x(3)\mathrm{Me} = x_{(3)}

The number in the middle of an ordered data set.

Legend:
  • x1 — Number 1
  • x2 — Number 2
  • x3 — Number 3
  • x4 — Number 4
  • x5 — Number 5

Mode of a sample (5 numbers)

Grade 9
Mo\mathrm{Mo}

The value that occurs most often in a data set.

Legend:
  • x1 — Number 1
  • x2 — Number 2
  • x3 — Number 3
  • x4 — Number 4
  • x5 — Number 5

Range of a sample (5 numbers)

Grade 9
R=xmax−xminR = x_{max} - x_{min}

Difference between the largest and smallest value in a data set.

Legend:
  • x1 — Number 1
  • x2 — Number 2
  • x3 — Number 3
  • x4 — Number 4
  • x5 — Number 5

Triangles

Perimeter of an arbitrary triangle

Grade 5
P=a+b+cP = a+b+c

Sum of the lengths of all three sides of a triangle.

abc
The perimeter is the sum of the three side lengths.
Legend:
  • a — Side a
  • b — Side b
  • c — Side c

Perimeter of an equilateral triangle

Grade 5
P=3aP = 3a

Sum of the three equal sides of an equilateral triangle.

aaa
Three equal sides — the perimeter is three times a side.
Legend:
  • a — Side a

Sum of the angles of a triangle

Grade 7
α+β+γ=180∘\alpha + \beta + \gamma = 180^\circ

Finds the third angle of a triangle from the two known angles.

αβγ
The sum of the three interior angles of any triangle is 180°.
Legend:
  • alpha — Angle α (deg)
  • beta — Angle β (deg)

Exterior angle of a triangle

Grade 7
α=β+γ\alpha = \beta + \gamma

An exterior angle equals the sum of the two interior angles not adjacent to it.

αβφ
The exterior angle φ equals the sum of the two non-adjacent interior angles.
Legend:
  • beta — Angle β (deg)
  • gamma — Angle γ (deg)

Triangle inequality

Grade 7
a<b+ca < b + c

Checks whether a triangle can exist for two given sides b and c.

cab
Any side is shorter than the sum of the other two.
Legend:
  • b — Side b
  • c — Side c

Perimeter of an isosceles triangle

Grade 7
P=2a+bP = 2a + b

Perimeter of a triangle with legs a and base b.

aab
The two legs are equal, b is the base.
Legend:
  • a — Leg a
  • b — Base b

Property of adjacent angles

Grade 7
α+β=180∘\alpha + \beta = 180^\circ

The sum of two adjacent angles equals 180°.

βα
Adjacent angles together form a straight angle of 180°.
Legend:
  • alpha — Angle α (deg)

Property of vertical angles

Grade 7
α=β\alpha = \beta

Vertical angles are equal to each other.

αα
Vertical angles (opposite each other) are equal.
Legend:
  • alpha — Angle α (deg)

Property of an isosceles triangle

Grade 7
β1=β2\beta_1 = \beta_2

The base angles of an isosceles triangle are equal.

β₁β₂
The base angles of an isosceles triangle are equal.
Legend:
  • alpha — Angle α (deg)

Pythagorean theorem

Grade 8
a2+b2=c2a^2 + b^2 = c^2

Finding the hypotenuse c of a right triangle from its two legs.

abc
c is the hypotenuse (opposite the right angle), a and b are the legs.
Legend:
  • a — Leg a
  • b — Leg b

Area of a triangle

Grade 8
S=12a⋅hS = \frac{1}{2} a \cdot h

The area of a triangle equals half the product of a side and its height.

ah
h is the height dropped onto side a at a right angle.
Legend:
  • a — Side a
  • h — Height h

Area of a right triangle

Grade 8
S=12a⋅bS = \frac{1}{2} a \cdot b

The area of a right triangle equals half the product of its legs.

ab
The legs a and b serve as the base and the height at the same time.
Legend:
  • a — Leg a
  • b — Leg b

Midsegment of a triangle

Grade 8
m=12am = \frac{1}{2} a

The midsegment equals half of the side it is parallel to.

am
The midsegment joins the midpoints of two sides and equals half the third.
Legend:
  • a — Parallel side a

Similarity ratio of triangles

Grade 8
k=aa′k = \frac{a}{a'}

Ratio of corresponding sides of similar triangles.

aa₁меньшийбольший
k is how many times larger one figure is than the other.
Legend:
  • a — Side of the original triangle (a)
  • aPrime — Side of the similar triangle (a')

Ratio of areas of similar figures

Grade 8
S1S2=k2\frac{S_1}{S_2} = k^2

The ratio of the areas of similar figures equals the square of the similarity ratio.

aa₁меньшийбольший
Areas are in the ratio of the square of the similarity ratio.
Legend:
  • k — Similarity ratio (k)

Thales' theorem (fourth proportional segment)

Grade 8
ab=cd⇒d=b⋅ca\frac{a}{b} = \frac{c}{d} \Rightarrow d = \frac{b \cdot c}{a}

Finds the fourth segment proportional to three known ones.

abcd
Parallel lines cut off proportional segments.
Legend:
  • a — Segment a
  • b — Segment b
  • c — Segment c

Area of an equilateral triangle

Grade 8
S=a234S = \frac{a^2\sqrt{3}}{4}

Area of a triangle with three equal sides.

aaaS
In an equilateral triangle all three sides are equal.
Legend:
  • a — Side a

Ratio of perimeters of similar figures

Grade 8
P1P2=k\frac{P_1}{P_2} = k

The ratio of the perimeters of similar figures equals the similarity ratio.

aa₁меньшийбольший
Perimeters are in the same ratio as the sides.
Legend:
  • k — Similarity ratio (k)

Triangle similarity test (SAS)

Grade 8
a1a2=b1b2⇒△∼△\frac{a_1}{a_2} = \frac{b_1}{b_2} \Rightarrow \triangle \sim \triangle

Checks the similarity of triangles from two sides and the angle between them (SAS).

αα
Two pairs of proportional sides and an equal angle between them.
Legend:
  • a1 — Side of the original triangle (a)
  • b1 — Side of the original triangle (b)
  • a2 — Side of the similar triangle (a')
  • b2 — Side of the similar triangle (b')

Law of cosines

Grade 9
a2=b2+c2−2bccos⁡αa^2 = b^2 + c^2 - 2bc \cos \alpha

Finds side a from two known sides and the angle between them.

αcba
a is the side opposite angle α.
Legend:
  • b — Side b
  • c — Side c
  • deg — Angle α (in degrees)

Law of sines

Grade 9
asin⁡A=bsin⁡B=csin⁡C=2R\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

Calculates the circumradius R of a triangle from side a and angle A.

R
R is the radius of the circle circumscribed around the triangle.
Legend:
  • a — Side a
  • degA — Angle A (in degrees)

Area of a triangle (via an angle)

Grade 9
S=12absin⁡γS = \frac{1}{2} a b \sin \gamma

Area of a triangle from two sides and the angle between them.

γab
γ is the angle between sides a and b.
Legend:
  • a — Side a
  • b — Side b
  • deg — Angle γ (in degrees)

Heron's formula

Grade 9
S=p(p−a)(p−b)(p−c),p=a+b+c2S = \sqrt{p(p-a)(p-b)(p-c)}, \quad p = \frac{a+b+c}{2}

Area of a triangle from its three known sides.

abc
Only three sides are needed — no height required.
Legend:
  • a — Side a
  • b — Side b
  • c — Side c

Area of a triangle (via circumradius)

Grade 9
S=abc4RS = \frac{abc}{4R}

Area of a triangle from its three sides and the circumradius.

R
R is the radius of the circle circumscribed around the triangle.
Legend:
  • a — Side a
  • b — Side b
  • c — Side c
  • R — Circumradius (R)

Area of a triangle (via inradius)

Grade 9
S=p⋅rS = p \cdot r

Area of a triangle from the semiperimeter and the inradius.

r
r is the radius of the circle inscribed in the triangle.
Legend:
  • p — Semiperimeter (p)
  • r — Inradius (r)

Perimeter of a triangle (via inradius and area)

Grade 9
P=2SrP = \frac{2S}{r}

Perimeter of a triangle from its area and the inradius.

rS
S is the area of the triangle, r is the inradius.
Legend:
  • S — Area (S)
  • r — Inradius (r)

Areas of figures

Area of a rectangle

Grade 5
S=a⋅bS = a \cdot b

Product of length and width.

Sab
The area is the product of two adjacent sides.
Legend:
  • a — Side a
  • b — Side b

Area of a square

Grade 5
S=a2S = a^2

The square of its side length.

Sa
The area of a square is its side multiplied by itself.
Legend:
  • a — Side a

Area of a parallelogram

Grade 8
S=a⋅haS = a \cdot h_a

Area of a parallelogram via a side and the height dropped onto it.

ah
h is the height drawn to side a.
Legend:
  • a — Side a
  • ha — Height h_a

Area of a rhombus (via diagonals)

Grade 8
S=12d1⋅d2S = \frac{1}{2} d_1 \cdot d_2

The area of a rhombus equals half the product of its diagonals.

d₁d₂
d₁ and d₂ are the diagonals, intersecting at a right angle.
Legend:
  • d1 — Diagonal d1
  • d2 — Diagonal d2

Diagonals of a parallelogram

Grade 8
OA=OC=d2OA = OC = \frac{d}{2}

The diagonals of a parallelogram bisect each other at their intersection point.

d₁d₂ab
The sum of the squared diagonals is related to the sides.
Legend:
  • d — Diagonal of the parallelogram (d)

Sum of angles on one side of a parallelogram

Grade 8
β=180∘−α\beta = 180^\circ - \alpha

The sum of the angles adjacent to one side of a parallelogram equals 180°.

αβab
Adjacent angles of a parallelogram add up to 180°.
Legend:
  • alpha — Angle α (deg)

Diagonal of a square

Grade 8
d=a2d = a\sqrt{2}

Diagonal of a square from its side length.

ad
The diagonal of a square joins opposite vertices.
Legend:
  • a — Side a

Diagonal of a rectangle

Grade 8
d=a2+b2d = \sqrt{a^2+b^2}

Diagonal of a rectangle from its length and width.

abd
The diagonal is found using the Pythagorean theorem.
Legend:
  • a — Side a
  • b — Side b

Area of a rhombus (via side and height)

Grade 8
S=a⋅hS = a \cdot h

Area of a rhombus as the product of a side and the height.

ha
h is the height dropped onto side a.
Legend:
  • a — Side a
  • h — Height h

Area of a circumscribed polygon

Grade 9
S=p⋅rS = p \cdot r

Area via the semiperimeter p and the inradius r.

ra
r is the inradius, p is the semiperimeter.
Legend:
  • p — Semiperimeter (p)
  • r — Inradius (r)

Area of a regular polygon

Grade 9
S=12PrS = \frac{1}{2} P r

Area of a regular polygon via the perimeter P and the inradius r.

ra
In a regular polygon all sides and angles are equal.
Legend:
  • P — Perimeter (P)
  • r — Inradius (r)

Area of a square (via diagonal)

Grade 9
S=d22S = \frac{d^2}{2}

Area of a square expressed through its diagonal.

Sd
d is the diagonal of the square, S is its area.
Legend:
  • d — Diagonal (d)

Area of a parallelogram (via sides and angle)

Grade 9
S=a⋅b⋅sin⁡αS = a \cdot b \cdot \sin \alpha

Area of a parallelogram from two sides and the angle between them.

αab
α is the angle between sides a and b.
Legend:
  • a — Side a
  • b — Side b
  • alpha — Angle α (in degrees)

Area of a parallelogram (via diagonals and angle)

Grade 9
S=12d1d2sin⁡γS = \frac{1}{2} d_1 d_2 \sin \gamma

Area of a parallelogram from its diagonals and the angle between them.

d₁d₂γ
γ is the angle between diagonals d₁ and d₂.
Legend:
  • d1 — Diagonal d1
  • d2 — Diagonal d2
  • gamma — Angle γ (in degrees)

Area of a rhombus (via side and angle)

Grade 9
S=a2⋅sin⁡αS = a^2 \cdot \sin \alpha

Area of a rhombus from a side and the angle between the sides.

αa
α is the angle between two sides of the rhombus.
Legend:
  • a — Side a
  • alpha — Angle α (in degrees)

Area of a regular polygon (via side)

Grade 9
S=na24tan⁡(180∘/n)S = \frac{n a^2}{4 \tan(180^\circ / n)}

Area of a regular n-gon from its side length, without using the inradius.

an сторін
a is the side, n is the number of sides of the polygon.
Legend:
  • n — Number of sides (n)
  • a — Side (a)

Area of an ellipse

Grade 9
S=πabS = \pi a b

Area of an ellipse from the lengths of its semi-major and semi-minor axes.

ab
a is the semi-major axis, b is the semi-minor axis of the ellipse.
Legend:
  • a — Semi-major axis (a)
  • b — Semi-minor axis (b)

Perimeters of figures

Perimeter of a rectangle

Grade 5
P=2⋅(a+b)P = 2 \cdot (a + b)

Sum of the lengths of all sides of a rectangle.

ab
The perimeter is the sum of the lengths of all four sides.
Legend:
  • a — Side a
  • b — Side b

Perimeter of a square

Grade 5
P=4⋅aP = 4 \cdot a

Sum of the four equal sides of a square.

aaaa
All four sides of a square are equal.
Legend:
  • a — Side a

Perimeter of an arbitrary polygon

Grade 5
P=a1+a2+a3+a4+a5P = a_1 + a_2 + a_3 + a_4 + a_5

The perimeter of any polygon equals the sum of the lengths of all its sides. Shown for a pentagon — for any other number of sides the sides are added the same way.

a₁a₂a₃a₄a₅
The perimeter is the sum of the lengths of all the sides, however many there are.
Legend:
  • a1 — Side a₁
  • a2 — Side a₂
  • a3 — Side a₃
  • a4 — Side a₄
  • a5 — Side a₅

Perimeter of a parallelogram

Grade 8
P=2(a+b)P = 2(a+b)

Sum of the lengths of all sides of a parallelogram.

ab
Opposite sides of a parallelogram are equal.
Legend:
  • a — Side a
  • b — Side b

Perimeter of a rhombus

Grade 8
P=4aP = 4a

Sum of the four equal sides of a rhombus.

aaaa
All four sides of a rhombus are equal.
Legend:
  • a — Side a

Perimeter of a regular polygon

Grade 8
P=n⋅aP = n \cdot a

Perimeter of a regular polygon as the number of sides times the side length.

a
n equal sides of length a.
Legend:
  • n — Number of sides (n)
  • a — Side (a)

Perimeter of a square (via diagonal)

Grade 8
P=22 dP = 2\sqrt{2}\, d

Perimeter of a square expressed through its diagonal.

dP = 4a
d is the diagonal of the square — the perimeter can be expressed through it too.
Legend:
  • d — Diagonal (d)

Trapezoid

Area of a trapezoid

Grade 8
S=a+b2⋅hS = \frac{a + b}{2} \cdot h

The area of a trapezoid equals the product of the half-sum of the bases and the height.

abh
a and b are the bases, h is the height between them.
Legend:
  • a — Base a
  • b — Base b
  • h — Height h

Midsegment of a trapezoid

Grade 8
m=a+b2m = \frac{a + b}{2}

The midsegment of a trapezoid equals the half-sum of its bases.

abm
m is the midsegment, joining the midpoints of the legs.
Legend:
  • a — Base a
  • b — Base b

Perimeter of a trapezoid

Grade 8
P=a+b+c+dP = a + b + c + d

Sum of the lengths of all four sides of a trapezoid.

cdab
The perimeter is the sum of the two bases and the two legs.
Legend:
  • a — Base a
  • b — Base b
  • c — Leg c
  • d — Leg d

Height of a trapezoid (via area)

Grade 8
h=2Sa+bh = \frac{2S}{a+b}

Height of a trapezoid expressed through its area and bases.

hab
The height is derived from the trapezoid's area formula.
Legend:
  • S — Area (S)
  • a — Base a
  • b — Base b

Segment between the midpoints of the diagonals

Grade 8
MN=∣a−b∣2MN = \frac{|a-b|}{2}

The segment joining the midpoints of a trapezoid's diagonals equals half the difference of its bases.

ab
The segment between the midpoints of the diagonals is half the difference of the bases.
Legend:
  • a — Base a
  • b — Base b

Sum of angles on one leg of a trapezoid

Grade 8
β=180∘−α\beta = 180^\circ - \alpha

The sum of the angles adjacent to one leg of a trapezoid equals 180°.

αβab
The angles on one leg add up to 180°.
Legend:
  • alpha — Angle α (deg)

Base of a trapezoid via the midsegment

Grade 8
a=2m−ba = 2m - b

A trapezoid's base expressed through the midsegment and the other base.

mab
The midsegment is the half-sum of the bases.
Legend:
  • m — Midsegment m
  • b — Base b

Area of a trapezoid (via midsegment and height)

Grade 8
S=m⋅hS = m \cdot h

Area of a trapezoid as the product of the midsegment and the height.

mh
m is the midsegment, h is the height of the trapezoid.
Legend:
  • m — Midsegment m
  • h — Height h

Perimeter of an isosceles trapezoid

Grade 8
P=a+b+2cP = a + b + 2c

Perimeter of an isosceles trapezoid, whose two legs are equal.

abcc
Both legs c are equal — the trapezoid is isosceles.
Legend:
  • a — Base a
  • b — Base b
  • c — Leg c

Sum of the squared diagonals of a trapezoid

Grade 9
d12+d22=2ab+c2+d2d_1^2 + d_2^2 = 2ab + c^2 + d^2

Relates a trapezoid's diagonals to the lengths of its bases and legs.

d₁d₂
d₁ and d₂ are the diagonals of the trapezoid.
Legend:
  • a — Base a
  • b — Base b
  • c — Leg c
  • d — Leg d

Base of a trapezoid via height and base angles

Grade 9
a=b+h⋅(ctg⁡α+ctg⁡β)a = b + h \cdot (\operatorname{ctg}\alpha + \operatorname{ctg}\beta)

The larger base of a trapezoid via the smaller base, the height and the angles at the lower base.

αβhab
The height and the base angles complete the larger base.
Legend:
  • b — Base b
  • h — Height h
  • alpha — Angle α (deg)
  • beta — Angle β (deg)

Base of a trapezoid via legs and angles

Grade 9
a=b+ccos⁡α+dcos⁡βa = b + c\cos\alpha + d\cos\beta

The larger base of a trapezoid via the smaller base, the legs and the angles at the lower base.

αβcdab
The legs and the base angles give the larger base.
Legend:
  • b — Base b
  • c — Leg c
  • d — Leg d
  • alpha — Angle α (deg)
  • beta — Angle β (deg)

Leg of a trapezoid via height and angle

Grade 9
c=hsin⁡αc = \frac{h}{\sin \alpha}

The leg of a trapezoid via the height and the adjacent angle at the base.

αchab
The leg is the hypotenuse of a right triangle with leg h.
Legend:
  • h — Height h
  • alpha — Angle α (deg)

Height of a trapezoid via leg and angle

Grade 9
h=c⋅sin⁡αh = c \cdot \sin \alpha

The height of a trapezoid via a leg and the adjacent angle at the base.

αch
α is the angle at the base, c is the leg.
Legend:
  • c — Leg c
  • alpha — Angle α (deg)

Area of a trapezoid (via diagonals and angle)

Grade 9
S=12d1d2sin⁡φS = \frac{1}{2} d_1 d_2 \sin \varphi

Area of a trapezoid from its diagonals and the angle between them.

d₁d₂φ
φ is the angle between the diagonals of the trapezoid.
Legend:
  • d1 — Diagonal d1
  • d2 — Diagonal d2
  • phi — Angle φ (in degrees)

Circle and Disk

Circumference (via diameter)

Grade 6
C=π⋅dC = \pi \cdot d

Calculating the circle's boundary length from its diameter.

d
d is the diameter — it passes through the centre from edge to edge.
Legend:
  • d — Diameter (d)

Circumference (via radius)

Grade 6
C=2πrC = 2 \pi r

Calculating the circle's boundary length from its radius.

r
r is the radius — from the centre to any point on the circle.
Legend:
  • r — Radius (r)

Area of a circle

Grade 6
S=πr2S = \pi r^2

Area of the region enclosed by a circle of radius r.

rS
S is the area of the whole shaded part of the circle.
Legend:
  • r — Radius (r)

Relationship between diameter and radius

Grade 6
d=2rd = 2r

The diameter of a circle equals twice its radius.

rrd
The diameter consists of two radii.
Legend:
  • r — Radius (r)

Area of a circle via diameter

Grade 6
S=πD24S = \frac{\pi D^2}{4}

Area of a circle expressed through its diameter.

DS
The same circle area, but expressed via the diameter.
Legend:
  • D — Diameter (D)

Property of two tangent segments from a point

Grade 8
AB=ACAB = AC

Two tangent segments to a circle drawn from the same point are equal.

ABC
Two tangent segments from point A are equal to each other.
Legend:
  • AB — Tangent segment AB

Equation of a circle centred at the origin

Grade 9
r2=x2+y2r^2 = x^2 + y^2

Radius of a circle centred at the origin passing through the point (x, y).

r(x, y)
The circle's centre coincides with the origin.
Legend:
  • x — Coordinate x
  • y — Coordinate y

Product of segments of two secants

Grade 9
AQ⋅BQ=CQ⋅DQAQ \cdot BQ = CQ \cdot DQ

If two secants are drawn from a point outside a circle, the products of their segments are equal.

QABCD
The products of the segments of both secants from point Q are equal.
Legend:
  • AQ — Segment AQ
  • BQ — Segment BQ
  • CQ — Segment CQ

A secant and a tangent from one point

Grade 9
AQ⋅BQ=CQ2AQ \cdot BQ = CQ^2

The square of the tangent segment equals the product of the secant's segments, from the same point.

QABC
QC is the tangent, QA and QB are the secant's segments.
Legend:
  • AQ — Segment AQ
  • BQ — Segment BQ

Chord length via the central angle

Grade 9
AB=2rsin⁡α2AB = 2r \sin \frac{\alpha}{2}

Length of a chord from the radius and the central angle.

αAB
α is the central angle subtending the chord AB.
Legend:
  • r — Radius (r)
  • alpha — Angle α (in degrees)

Chord length via the inscribed angle

Grade 9
AB=2rsin⁡αAB = 2r \sin \alpha

Length of a chord from the radius and the inscribed angle.

αAB
α is the inscribed angle subtending the chord AB.
Legend:
  • r — Radius (r)
  • alpha — Angle α (in degrees)

Length of a circular arc

Grade 9
l=πrα180∘l = \frac{\pi r \alpha}{180^\circ}

Calculates the length of an arc with central angle α.

αL
L is the length of the highlighted arc, α is its central angle.
Legend:
  • r — Radius (r)
  • alpha — Angle α (in degrees)

Area of a circular sector

Grade 9
S=πr2α360∘S = \frac{\pi r^2 \alpha}{360^\circ}

Area of a sector with central angle α.

αS
A sector is the part of a circle between two radii.
Legend:
  • r — Radius (r)
  • alpha — Angle α (in degrees)

Area of a circular segment

Grade 9
S=R22(θ−sin⁡θ)S = \frac{R^2}{2} (\theta - \sin \theta)

Area of a circular segment (the part of a circle cut off by a chord); the angle θ is given in radians.

Sθ
S is the segment cut off by the chord, θ is the central angle.
Legend:
  • R — Radius (R)
  • theta — Angle θ (in radians)

Area of an annulus (ring)

Grade 9
S=π(R2−r2)S = \pi (R^2 - r^2)

Area of the ring between two concentric circles of radii R and r.

Rr
The ring is the region between circles of radii R and r.
Legend:
  • R — Outer radius (R)
  • r — Inner radius (r)

Sequences

nth term of an arithmetic sequence

Grade 9
an=a1+(n−1)da_n = a_1 + (n - 1)d

Lets you find any term of an arithmetic sequence.

Legend:
  • a1 — First term (a1)
  • n — Element number (n)
  • d — Common difference (d)

Sum of the first n terms of an arithmetic sequence

Grade 9
Sn=a1+an2⋅nS_n = \frac{a_1 + a_n}{2} \cdot n

Calculates the sum from the 1st to the nth term of an arithmetic sequence.

Legend:
  • a1 — First term (a1)
  • an — nth term (an)
  • n — Number of elements (n)

nth term of a geometric sequence

Grade 9
bn=b1⋅qn−1b_n = b_1 \cdot q^{n-1}

Lets you find any term of a geometric sequence.

Legend:
  • b1 — First term (b1)
  • q — Common ratio (q)
  • n — Element number (n)

Sum of the first n terms of a geometric sequence

Grade 9
Sn=b1(qn−1)q−1(q≠1)S_n = \frac{b_1 (q^n - 1)}{q - 1} \quad (q \neq 1)

Calculates the sum of the first n terms of a geometric sequence when q ≠ 1.

Legend:
  • b1 — First term (b1)
  • q — Common ratio q (q≠1)
  • n — Count (n)

Sum of n terms of an arithmetic sequence (via first term and difference)

Grade 9
Sn=2a1+(n−1)d2⋅nS_n = \frac{2a_1 + (n-1)d}{2} \cdot n

Calculates the sum of the first n terms of an arithmetic sequence from the first term and the common difference.

Legend:
  • a1 — First term (a1)
  • n — Count (n)
  • d — Common difference (d)

Characteristic property of a geometric sequence

Grade 9
bn2=bn+1⋅bn−1b_n^2 = b_{n+1} \cdot b_{n-1}

The square of a term of a geometric sequence equals the product of its neighbouring terms.

Legend:
  • bnext — Next term (b_{n+1})
  • bprev — Previous term (b_{n-1})

Characteristic property of an arithmetic sequence

Grade 9
an=an−1+an+12a_n = \frac{a_{n-1}+a_{n+1}}{2}

Each term of an arithmetic sequence equals the arithmetic mean of its neighbours.

Legend:
  • aprev — Previous term (a_{n-1})
  • anext — Next term (a_{n+1})

Common difference of an arithmetic sequence (via two arbitrary terms)

Grade 9
d=an+1−an=an−akn−kd = a_{n+1} - a_n = \frac{a_n - a_k}{n - k}

The common difference of an arithmetic sequence from any two of its terms and their indices.

a_ka_na_{n+1}d
d is the step between neighbouring terms, the same throughout the sequence.
Legend:
  • an — Term a_n
  • ak — Term a_k
  • n — Index n
  • k — Index k

nth term of an arithmetic sequence (via an arbitrary kth term)

Grade 9
an=ak+(n−k)da_n = a_k + (n - k)d

Any term of an arithmetic sequence can be found not only from the first term, but from any other known term.

a_ka_n(n − k) шагов по d
From the kth to the nth term there are (n − k) equal steps of d.
Legend:
  • ak — Term a_k
  • n — Index n
  • k — Index k
  • d — Common difference (d)

nth term of a geometric sequence (via an arbitrary kth term)

Grade 9
bn=bk⋅qn−kb_n = b_k \cdot q^{n-k}

Any term of a geometric sequence can be found not only from the first term, but from any other known term.

b_kb_nумножение на q, (n − k) раз
From the kth to the nth term we multiply by q exactly (n − k) times.
Legend:
  • bk — Term b_k
  • n — Index n
  • k — Index k
  • q — Common ratio q

Sum of an infinite geometric series

Grade 10
S=b11−q, ∣q∣<1S = \frac{b_1}{1-q},\ |q|<1

Sum of all terms of an infinitely decreasing geometric progression (|q| < 1).

Legend:
  • b1 — First term (b1)
  • q — Common ratio (q)

Combinatorics

Number of permutations

Grade 9
Pn=n!P_n = n!

Number of ways to arrange n elements (n factorial).

Legend:
  • n — Number of elements n (integer ≥ 0)

Number of combinations C(n,k)

Grade 9
Cnk=n!k!(n−k)!C_n^k = \frac{n!}{k!(n-k)!}

Number of ways to choose k elements out of n, order not counted.

Legend:
  • n — Number of elements (n)
  • k — Number chosen (k)

Number of permutations A(n,k)

Grade 9
Ank=n!(n−k)!A_n^k = \frac{n!}{(n-k)!}

Number of ways to choose and order k elements out of n.

Legend:
  • n — Number of elements (n)
  • k — Number chosen (k)

Classical probability of an event

Grade 9
P(A)=mnP(A) = \frac{m}{n}

Probability of an event as the ratio of favourable outcomes to all outcomes.

Legend:
  • m — Favourable outcomes (m)
  • n — Total outcomes (n)

Probability of the complementary event

Grade 9
P(Aˉ)=1−P(A)P(\bar A) = 1 - P(A)

Probability that event A does not occur.

Legend:
  • p — Probability of event P(A)

Inclusion–exclusion principle (for two sets)

Grade 9
∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A \cup B| = |A| + |B| - |A \cap B|

Determines the number of elements in the union of two finite sets.

Legend:
  • A — Cardinality of set |A|
  • B — Cardinality of set |B|
  • AB — Cardinality |A∩B|

Inclusion–exclusion principle (for three sets)

Grade 9
∣A∪B∪C∣=∣A∣+∣B∣+∣C∣−∣A∩B∣−∣A∩C∣−∣B∩C∣+∣A∩B∩C∣|A \cup B \cup C| = |A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|

Determines the number of elements in the union of three finite sets.

Legend:
  • A — Cardinality of set |A|
  • B — Cardinality of set |B|
  • C — Cardinality of set |C|
  • AB — Cardinality |A∩B|
  • AC — Cardinality |A∩C|
  • BC — Cardinality |B∩C|
  • ABC — Cardinality |A∩B∩C|

Probability of independent events

Grade 11
P(A∩B)=P(A)⋅P(B)P(A \cap B) = P(A) \cdot P(B)

Probability that two independent events both occur.

Legend:
  • pa — Probability of event P(A)
  • pb — Probability of event P(B)

Bernoulli's formula

Grade 11
Pn(k)=Cnk⋅pk⋅(1−p)n−kP_n(k) = C_n^k \cdot p^k \cdot (1-p)^{n-k}

Probability of exactly k successes in n independent trials with success probability p.

Legend:
  • n — Number of elements (n)
  • k — Number chosen (k)
  • p — Probability of success (p)

Powers and Logarithms

Finding 1% of a number

Grade 5
A100\frac{A}{100}

One hundredth of a given number A.

Legend:
  • A — Number A

Finding p% of a number A

Grade 5
B=A⋅p100B = A \cdot \frac{p}{100}

The value of a given percentage p of a number A.

Legend:
  • A — Number A
  • p — Percentage p

Finding a number from its percentage

Grade 5
A=B⋅100pA = B \cdot \frac{100}{p}

The number A, given that B is p% of it.

Legend:
  • B — Known part (B)
  • p — Percentage p

Multiplying powers

Grade 7
am⋅an=am+na^m \cdot a^n = a^{m+n}

When multiplying powers with the same base, the exponents are added.

Legend:
  • a — Base a
  • m — Exponent m
  • n — Exponent n

Dividing powers

Grade 7
am:an=am−na^m : a^n = a^{m-n}

When dividing powers with the same base, the exponents are subtracted.

Legend:
  • a — Base a
  • m — Exponent m
  • n — Exponent n

Power of a power

Grade 7
(am)n=am⋅n(a^m)^n = a^{m \cdot n}

When raising a power to a power, the exponents are multiplied.

Legend:
  • a — Base a
  • m — Exponent m
  • n — Exponent n

Power of a product

Grade 7
(a⋅b)n=an⋅bn(a \cdot b)^n = a^n \cdot b^n

Each factor is raised to the given power.

Legend:
  • a — Factor a
  • b — Factor b
  • n — Exponent n

Zero exponent

Grade 7
a0=1(a≠0)a^0 = 1 \quad (a \neq 0)

Any nonzero number raised to the power of 0 equals 1.

Legend:
  • a — Base a (a≠0)

Properties of a square root (multiplication)

Grade 8
a⋅b=a⋅b(a,b≥0)\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \quad (a, b \ge 0)

The root of a product of non-negative numbers equals the product of their roots.

Legend:
  • a — Number a (a≥0)
  • b — Number b (b≥0)

Properties of a square root (division)

Grade 8
ab=ab(a≥0,b>0)\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \quad (a \ge 0, b > 0)

The root of a quotient equals the quotient of the roots of the numerator and denominator.

Legend:
  • a — Number a (a≥0)
  • b — Number b (b>0)

Square of a root

Grade 8
(a)2=a(a≥0)(\sqrt{a})^2 = a \quad (a \ge 0)

Squaring the arithmetic square root.

Legend:
  • a — Number a (a≥0)

Root of a square

Grade 8
a2=∣a∣\sqrt{a^2} = |a|

The square root of the square of a number equals its absolute value.

Legend:
  • a — Number a

Negative exponent

Grade 8
a−n=1an(a≠0)a^{-n} = \frac{1}{a^n} \quad (a \neq 0)

Replacing a negative exponent with the reciprocal value.

Legend:
  • a — Base a (a≠0)
  • n — Exponent n

Rational exponent

Grade 10
amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}

Expressing a power with a fractional exponent as a root.

Legend:
  • a — Base a
  • m — Exponent m
  • n — Root index n (n≠0)

Definition of a logarithm

Grade 10
log⁡ab=c  ⟺  ac=b(a>0,a≠1,b>0)\log_a b = c \iff a^c = b \quad (a>0, a \neq 1, b>0)

Finds the exponent c to which base a must be raised to get b.

Legend:
  • a — Base a (a>0, a≠1)
  • b — Argument b (b>0)

Fundamental logarithmic identity

Grade 10
alog⁡ab=ba^{\log_a b} = b

Raising base a to the power log_a(b) gives the argument b.

Legend:
  • b — Number b (b>0)

Properties of logarithms (sum)

Grade 10
log⁡ax+log⁡ay=log⁡a(x⋅y)\log_a x + \log_a y = \log_a (x \cdot y)

The sum of logarithms equals the logarithm of the product of the arguments.

Legend:
  • a — Base a
  • x — Argument x
  • y — Argument y

Properties of logarithms (difference)

Grade 10
log⁡ax−log⁡ay=log⁡a(xy)\log_a x - \log_a y = \log_a \left(\frac{x}{y}\right)

The difference of logarithms equals the logarithm of the quotient of the arguments.

Legend:
  • a — Base a
  • x — Argument x
  • y — Argument y

Taking an exponent out of a logarithm

Grade 10
log⁡a(xk)=k⋅log⁡ax\log_a (x^k) = k \cdot \log_a x

The exponent k is moved out as a factor in front of the logarithm.

Legend:
  • a — Base a
  • x — Argument x
  • k — Exponent k

Change-of-base formula

Grade 10
log⁡ab=log⁡cblog⁡ca\log_a b = \frac{\log_c b}{\log_c a}

Converting a logarithm with base a to base c.

Legend:
  • a — Old base a
  • b — Argument b
  • c — New base c

Logarithm of 1

Grade 10
log⁡a1=0\log_a 1 = 0

The logarithm of 1 to any base is always 0.

Legend:
  • a — Base a (a>0, a≠1)

Logarithm of a number equal to the base

Grade 10
log⁡aa=1\log_a a = 1

The logarithm of a number to an equal base is always 1.

Legend:
  • a — Base a (a>0, a≠1)

Logarithm with the base raised to a power

Grade 11
log⁡anx=1nlog⁡ax\log_{a^n} x = \frac{1}{n}\log_a x

Lets you move the exponent of the base out from under the logarithm (for n ≠ 0).

Legend:
  • n — Exponent n
  • a — Base a (a>0, a≠1)
  • x — Argument x

Reciprocal logarithm bases

Grade 11
log⁡ax=1log⁡xa\log_a x = \frac{1}{\log_x a}

Relates the logarithm with base a to the logarithm with the reciprocal base.

Legend:
  • a — Base a (a>0, a≠1)
  • x — Argument x

Trigonometry

Sine of an acute angle (Sin)

Grade 8
sin⁡α=ac\sin \alpha = \frac{a}{c}

Ratio of the opposite leg to the hypotenuse.

αac
a is the opposite leg, c is the hypotenuse.
Legend:
  • opp — Opposite leg
  • hyp — Hypotenuse

Cosine of an acute angle (Cos)

Grade 8
cos⁡α=bc\cos \alpha = \frac{b}{c}

Ratio of the adjacent leg to the hypotenuse.

αbc
b is the adjacent leg, c is the hypotenuse.
Legend:
  • adj — Adjacent leg
  • hyp — Hypotenuse

Tangent of an acute angle (Tan)

Grade 8
tg⁡α=ab\operatorname{tg} \alpha = \frac{a}{b}

Ratio of the opposite leg to the adjacent leg.

αab
a is the opposite leg, b is the adjacent leg.
Legend:
  • opp — Opposite leg
  • adj — Adjacent leg

Fundamental trigonometric identity

Grade 9
sin⁡2α+cos⁡2α=1\sin^2 \alpha + \cos^2 \alpha = 1

Calculates |sin α| from a known cos α.

sin αcos α1
Follows from the Pythagorean theorem in the unit circle.
Legend:
  • cos — Cosine (cos α)

Relation between tangent, sine and cosine

Grade 9
tg⁡α=sin⁡αcos⁡α\operatorname{tg} \alpha = \frac{\sin \alpha}{\cos \alpha}

Calculating tan α as the ratio of sin α to cos α.

αsin αcos α
Tangent is the ratio of sine to cosine of the same angle.
Legend:
  • sin — Sine (sin α)
  • cos — Cosine (cos α)

Cotangent of an acute angle

Grade 9
ctg⁡ α=ba\operatorname{ctg}\,\alpha = \frac{b}{a}

Ratio of the adjacent leg to the opposite leg.

αab
Cotangent is the adjacent leg divided by the opposite leg.
Legend:
  • adj — Adjacent leg
  • opp — Opposite leg

Table of sine values

Grade 9

Values of sin α for the standard angles from 0° to 360°.

α0°30°45°60°90°120°180°270°360°
sin α01/2√2/2√3/21√3/20-10

Table of cosine values

Grade 9

Values of cos α for the standard angles from 0° to 360°.

α0°30°45°60°90°120°180°270°360°
cos α1√3/2√2/21/20-1/2-101

Table of tangent values

Grade 9

Values of tan α for the standard angles from 0° to 360°.

α0°30°45°60°90°120°180°270°360°
tg α01/√31√3—-√30—0

Table of cotangent values

Grade 9

Values of cot α for the standard angles from 0° to 360°.

α0°30°45°60°90°120°180°270°360°
ctg α—√311/√30-1/√3—0—

Table of angles in radians

Grade 9

Correspondence between standard angles in degrees and radians.

α0°30°45°60°90°120°180°270°360°
α (рад)0π/6π/4π/3π/22π/3π3π/22π

Sine of a sum and a difference

Grade 10
sin⁡(α±β)=sin⁡αcos⁡β±cos⁡αsin⁡β\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta

Calculates sin(α + β) and sin(α − β).

Legend:
  • degA — Angle α (deg)
  • degB — Angle β (deg)

Cosine of a sum and a difference

Grade 10
cos⁡(α±β)=cos⁡αcos⁡β∓sin⁡αsin⁡β\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta

Calculates cos(α + β) and cos(α − β).

Legend:
  • degA — Angle α (deg)
  • degB — Angle β (deg)

Sine of a double angle

Grade 10
sin⁡2α=2sin⁡αcos⁡α\sin 2\alpha = 2 \sin \alpha \cos \alpha

Formula for expanding the sine of a double angle.

Legend:
  • deg — Angle α (in degrees)

Cosine of a double angle

Grade 10
cos⁡2α=cos⁡2α−sin⁡2α\cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha

Formula for expanding the cosine of a double angle.

Legend:
  • deg — Angle α (in degrees)

Tangent of a double angle

Grade 10
tg⁡2α=2tg⁡α1−tg⁡2α\operatorname{tg} 2\alpha = \frac{2 \operatorname{tg} \alpha}{1 - \operatorname{tg}^2 \alpha}

Formula for the tangent of a double angle.

Legend:
  • deg — Angle α (in degrees)

Converting degrees to radians

Grade 10
rad=α⋅π180\mathrm{rad} = \alpha \cdot \frac{\pi}{180}

Converts an angle from degrees to radians.

α°рад
180° corresponds to π radians.
Legend:
  • alpha — Angle α (deg)

Converting radians to degrees

Grade 10
α=rad⋅180π\alpha = \mathrm{rad} \cdot \frac{180}{\pi}

Converts an angle from radians to degrees.

рад°
The reverse conversion: from radians to degrees.
Legend:
  • rad — Angle in radians (rad)

Derivatives and Integrals

Derivative of a constant

Grade 10
(c)′=0(c)' = 0

The derivative of a constant is always zero.

Legend:
  • c — Number c

Derivative of a power function

Grade 11
(xn)′=n⋅xn−1(x^n)' = n \cdot x^{n-1}

Value of the derivative of x^n at a given point x.

Legend:
  • x — Value x
  • n — Exponent n

Derivatives of trigonometric functions

Grade 11
(sin⁡x)′=cos⁡x,(cos⁡x)′=−sin⁡x(\sin x)' = \cos x, \quad (\cos x)' = -\sin x

Calculates the values of (sin x)' and (cos x)' at a point x.

Legend:
  • deg — Angle x (in degrees)

Derivative of an exponential function

Grade 11
(ax)′=axln⁡a,(ex)′=ex(a^x)' = a^x \ln a, \quad (e^x)' = e^x

Value of the derivative of a^x at a point x.

Legend:
  • a — Base a (a>0)
  • x — Exponent x

Derivative of a product

Grade 11
(u⋅v)′=u′v+uv′(u \cdot v)' = u'v + uv'

Differentiation rule for the product of two functions u and v.

Legend:
  • u — First function
  • u' — Derivative of the first function
  • v — Second function
  • v' — Derivative of the second function

Derivative of a quotient

Grade 11
(uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}

Differentiation rule for the quotient of two functions u and v.

Legend:
  • u — First function
  • u' — Derivative of the first function
  • v — Second function
  • v' — Derivative of the second function

Newton–Leibniz formula (integral)

Grade 11
∫abf(x)dx=F(b)−F(a)\int_a^b f(x)dx = F(b) - F(a)

Calculates a definite integral using values of the antiderivative.

Legend:
  • F(b) — Value of the antiderivative at point b
  • F(a) — Value of the antiderivative at point a

Chain rule (derivative of a composite function)

Grade 11
(f(g(x)))′=f′(g(x))⋅g′(x)(f(g(x)))' = f'(g(x)) \cdot g'(x)

Differentiation rule for a composite function (the chain rule).

Legend:
  • fu — Value f'(u)
  • gx — Value g'(x)

Derivative of a sum of functions

Grade 11
(u+v)′=u′+v′(u+v)' = u' + v'

The derivative of a sum of functions equals the sum of their derivatives.

Legend:
  • up — Derivative u'
  • vp — Derivative v'

Equation of a tangent line to a function's graph

Grade 11
y=f(a)+f′(a)(x−a)y = f(a) + f'(a)(x-a)

Value of the tangent line to a function's graph at a point x.

Legend:
  • fa — Function value f(a)
  • fpa — Derivative value f'(a)
  • a0 — Coordinate x0
  • x — Argument x

Antiderivative of a power function

Grade 11
F(x)=xn+1n+1F(x) = \frac{x^{n+1}}{n+1}

Antiderivative of the power function xⁿ for n ≠ −1.

Legend:
  • x — Value x
  • n — Exponent n

Derivative of x

Grade 11
x′=1x' = 1

The derivative of the variable x with respect to itself is always 1.

Legend:
  • x — Value x

Derivative of 1/x

Grade 11
(1x)′=−1x2\left(\frac{1}{x}\right)' = -\frac{1}{x^2}

Derivative of the inverse-proportionality function at a point x.

Legend:
  • x — Value x

Derivative of a square root

Grade 11
(x)′=12x(\sqrt{x})' = \frac{1}{2\sqrt{x}}

Derivative of the square root function at a point x (x > 0).

Legend:
  • x — Value x

Derivative of tangent

Grade 11
(tg⁡x)′=1cos⁡2x(\operatorname{tg} x)' = \frac{1}{\cos^2 x}

Derivative of the tangent function at a point x.

Legend:
  • deg — Angle x (in degrees)

Derivative of cotangent

Grade 11
(ctg⁡x)′=−1sin⁡2x(\operatorname{ctg} x)' = -\frac{1}{\sin^2 x}

Derivative of the cotangent function at a point x.

Legend:
  • deg — Angle x (in degrees)

Derivative of a logarithm with base a

Grade 11
(log⁡ax)′=1xln⁡a(\log_a x)' = \frac{1}{x \ln a}

Derivative of a logarithmic function with an arbitrary base a.

Legend:
  • x — Value x
  • a — Base a (a>0, a≠1)

Derivative of the natural logarithm

Grade 11
(ln⁡x)′=1x(\ln x)' = \frac{1}{x}

Derivative of the natural logarithm at a point x.

Legend:
  • x — Value x

Antiderivative of a constant

Grade 11
F(x)=kx+CF(x) = kx + C

Antiderivative of a constant k.

Legend:
  • k — Coefficient k
  • x — Value x

Antiderivative of 1/x

Grade 11
F(x)=ln⁡∣x∣+CF(x) = \ln|x| + C

Antiderivative of the inverse-proportionality function.

Legend:
  • x — Value x

Antiderivative of 1/√x

Grade 11
F(x)=2x+CF(x) = 2\sqrt{x} + C

Antiderivative of the reciprocal of the square root of x.

Legend:
  • x — Value x

Antiderivative of sine

Grade 11
F(x)=−cos⁡x+CF(x) = -\cos x + C

Antiderivative of the sine function.

Legend:
  • deg — Angle x (in degrees)

Antiderivative of cosine

Grade 11
F(x)=sin⁡x+CF(x) = \sin x + C

Antiderivative of the cosine function.

Legend:
  • deg — Angle x (in degrees)

Antiderivative of 1/cos²x

Grade 11
F(x)=tg⁡x+CF(x) = \operatorname{tg} x + C

Antiderivative of the reciprocal of cosine squared.

Legend:
  • deg — Angle x (in degrees)

Antiderivative of 1/sin²x

Grade 11
F(x)=−ctg⁡x+CF(x) = -\operatorname{ctg} x + C

Antiderivative of the reciprocal of sine squared.

Legend:
  • deg — Angle x (in degrees)

Antiderivative of an exponential function

Grade 11
F(x)=axln⁡a+CF(x) = \frac{a^x}{\ln a} + C

Antiderivative of an exponential function with an arbitrary base a.

Legend:
  • a — Base a (a>0, a≠1)
  • x — Value x

Antiderivative of eˣ

Grade 11
F(x)=ex+CF(x) = e^x + C

Antiderivative of the exponential function eˣ.

Legend:
  • x — Value x

Vectors

Distance between two points

Grade 8
d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Length of the segment between two points on the coordinate plane.

ABd
d is the distance between points A and B in the plane.
Legend:
  • x1 — x-coordinate of the first point
  • y1 — y-coordinate of the first point
  • x2 — x-coordinate of the second point
  • y2 — y-coordinate of the second point

Midpoint of a segment (coordinates)

Grade 8
M(x1+x22, y1+y22)M\left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)

Coordinates of the point bisecting a segment.

ABM
M is the midpoint of segment AB.
Legend:
  • x1 — x-coordinate of the first point
  • y1 — y-coordinate of the first point
  • x2 — x-coordinate of the second point
  • y2 — y-coordinate of the second point

Dot product of vectors

Grade 9
a⃗⋅b⃗=x1x2+y1y2\vec{a} \cdot \vec{b} = x_1 x_2 + y_1 y_2

Dot product of two 2D vectors from their coordinates.

ab
The dot product is calculated from the vectors' coordinates.
Legend:
  • x1 — x-coordinate of the first vector
  • y1 — y-coordinate of the first vector
  • x2 — x-coordinate of the second vector
  • y2 — y-coordinate of the second vector

Length of a vector from its coordinates

Grade 9
∣a⃗∣=x2+y2|\vec{a}| = \sqrt{x^2 + y^2}

Length (magnitude) of a vector in the plane.

xy
The vector's length follows from the Pythagorean theorem on its coordinates.
Legend:
  • x — Coordinate x
  • y — Coordinate y

Sum of vectors by coordinates

Grade 9
a⃗+b⃗=(ax+bx, ay+by)\vec{a}+\vec{b} = (a_x+b_x,\ a_y+b_y)

Adding vectors coordinate-wise.

aba+b
Vectors are added by the 'triangle rule'.
Legend:
  • a_x — x-coordinate of vector a
  • a_y — y-coordinate of vector a
  • b_x — x-coordinate of vector b
  • b_y — y-coordinate of vector b

Cosine of the angle between vectors (2D)

Grade 9
cos⁡φ=a⃗⋅b⃗∣a⃗∣∣b⃗∣\cos\varphi = \frac{\vec a \cdot \vec b}{|\vec a||\vec b|}

Cosine of the angle between two vectors in the plane from their coordinates.

α
Cosine of the angle between vectors in the plane.
Legend:
  • a_x — x-coordinate of vector a
  • a_y — y-coordinate of vector a
  • b_x — x-coordinate of vector b
  • b_y — y-coordinate of vector b

Slope of a line through two points

Grade 9
k=y2−y1x2−x1k = \frac{y_2-y_1}{x_2-x_1}

Slope of the line passing through two given points.

ΔxΔy
k is the ratio of the change in y to the change in x.
Legend:
  • x1 — x-coordinate of the first point
  • y1 — y-coordinate of the first point
  • x2 — x-coordinate of the second point
  • y2 — y-coordinate of the second point

Distance from a point to a line

Grade 9
d=∣Ax0+By0+C∣A2+B2d = \frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}

Shortest distance from a point to a line given by Ax + By + C = 0.

Md
d is the shortest (perpendicular) distance from the point to the line.
Legend:
  • A — Coefficient a
  • B — Coefficient b
  • C — Coefficient c
  • x0 — Coordinate x0
  • y0 — Coordinate y0

Radius of a circle from its equation

Grade 9
r=(x−a)2+(y−b)2r = \sqrt{(x-a)^2+(y-b)^2}

Radius of a circle centred at (a, b) passing through the point (x, y).

r(a, b)(x, y)
The circle's centre is at (a, b), the point (x, y) lies on it.
Legend:
  • x0 — Coordinate x0
  • y0 — Coordinate y0
  • x — Coordinate x
  • y — Coordinate y

Difference of vectors by coordinates

Grade 9
a⃗−b⃗=(ax−bx, ay−by)\vec a-\vec b = (a_x-b_x,\ a_y-b_y)

Subtracting vectors coordinate-wise.

aba−b
The vector difference points from the tip of b to the tip of a.
Legend:
  • ax — a_x
  • ay — a_y
  • bx — b_x
  • by — b_y

Dot product of vectors (via magnitudes and angle)

Grade 9
A⃗⋅B⃗=∣a⃗∣⋅∣b⃗∣⋅cos⁡α\vec{A} \cdot \vec{B} = |\vec a| \cdot |\vec b| \cdot \cos\alpha

Dot product of vectors through their lengths and the angle between them.

αab
α is the angle between vectors a and b.
Legend:
  • a — Magnitude of vector a (|a|)
  • b — Magnitude of vector b (|b|)
  • alpha — Angle α (deg)

Angle between vectors in space

Grade 11
cos⁡θ=x1x2+y1y2+z1z2∣a⃗∣⋅∣b⃗∣\cos \theta = \frac{x_1 x_2 + y_1 y_2 + z_1 z_2}{|\vec{a}| \cdot |\vec{b}|}

Cosine of the angle between two 3D vectors from their coordinates.

α
The angle between vectors in three-dimensional space.
Legend:
  • a_x — x-coordinate of the first vector
  • a_y — y-coordinate of the first vector
  • a_z — z-coordinate of the first vector
  • b_x — x-coordinate of the second vector
  • b_y — y-coordinate of the second vector
  • b_z — z-coordinate of the second vector

Length of a vector in space

Grade 11
∣a⃗∣=x2+y2+z2|\vec a| = \sqrt{x^2+y^2+z^2}

Length (magnitude) of a vector in 3D space from its coordinates.

xyz
In space, a third coordinate z is added.
Legend:
  • x — a_x
  • y — a_y
  • z — a_z

Distance between two points in space

Grade 11
AB=(x2−x1)2+(y2−y1)2+(z2−z1)2AB = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}

Length of the segment between two points in 3D space.

ABd
The distance between points in three-dimensional space.
Legend:
  • x1 — x-coordinate of the first point
  • y1 — y-coordinate of the first point
  • z1 — z-coordinate of the first point
  • x2 — x-coordinate of the second point
  • y2 — y-coordinate of the second point
  • z2 — z-coordinate of the second point

Solid geometry

Volume of a rectangular box

Grade 5
V=a⋅b⋅cV = a \cdot b \cdot c

Product of the three dimensions: length, width and height.

acb
a, b, c are the three dimensions of the box.
Legend:
  • a — Length (a)
  • b — Width (b)
  • c — Height (c)

Surface area of a rectangular box

Grade 5
S=2⋅(ab+bc+ac)S = 2 \cdot (ab + bc + ac)

Sum of the areas of all six faces of the box.

acb
The areas of all six faces are added together.
Legend:
  • a — Edge a
  • b — Edge b
  • c — Edge c

Sum of edge lengths of a rectangular box

Grade 5
L=4⋅(a+b+c)L = 4 \cdot (a + b + c)

Sum of the lengths of all 12 edges of a rectangular box.

acb
A rectangular box has 4 edges of each dimension.
Legend:
  • a — Edge a
  • b — Edge b
  • c — Edge c

Volume of a cube

Grade 5
V=a3V = a^3

The cube of the edge length.

aaa
All edges of a cube are equal.
Legend:
  • a — Edge a

Surface area of a cube

Grade 5
S=6⋅a2S = 6 \cdot a^2

Sum of the areas of the six equal square faces.

aaa
A cube has 6 identical square faces.
Legend:
  • a — Edge a

Lateral surface area of a rectangular box

Grade 5
S=2c(a+b)S = 2c(a+b)

Lateral surface area of a rectangular box from its three dimensions.

acb
Lateral surface = the four vertical faces.
Legend:
  • a — Edge a
  • b — Edge b
  • c — Edge c

Sum of edge lengths of a cube

Grade 5
L=12aL = 12a

Sum of the lengths of all 12 edges of a cube.

a
All 12 edges of the cube have the same length a — highlighted in green.
Legend:
  • a — Edge a

Volume of a prism

Grade 10
V=S⋅hV = S \cdot h

The volume of a prism equals the base area times the height.

hS
S is the base area, h is the height of the prism.
Legend:
  • S — Base area (S)
  • h — Height (h)

Volume of a pyramid

Grade 10
V=13S⋅hV = \frac{1}{3} S \cdot h

The volume of a pyramid equals one third of the base area times the height.

hS
S is the base area, h is the height of the pyramid.
Legend:
  • S — Base area (S)
  • h — Height (h)

Lateral surface area of a prism

Grade 10
Sb=P⋅hS_b = P \cdot h

Lateral surface area of a right prism via the base perimeter and the height.

hS
Lateral surface = all the side faces of the prism.
Legend:
  • P — Perimeter (P)
  • h — Height (h)

Volume of a cylinder

Grade 11
V=πr2hV = \pi r^2 h

Volume of a cylinder with base radius r and height h.

rh
r is the base radius, h is the height of the cylinder.
Legend:
  • r — Radius (r)
  • h — Height (h)

Lateral surface area of a cylinder

Grade 11
S=2πrhS = 2 \pi r h

Lateral surface area of a right circular cylinder.

rh
The lateral surface unrolls into a rectangle.
Legend:
  • r — Radius (r)
  • h — Height (h)

Volume of a cone

Grade 11
V=13πr2hV = \frac{1}{3} \pi r^2 h

Volume of a right circular cone.

rh
r is the base radius, h is the height of the cone.
Legend:
  • r — Radius (r)
  • h — Height (h)

Lateral surface area of a cone

Grade 11
S=πrlS = \pi r l

Lateral surface area of a cone via radius r and slant height l.

rl
l is the slant height of the cone, r is the base radius.
Legend:
  • r — Radius (r)
  • l — Slant height (l)

Volume of a sphere

Grade 11
V=43πR3V = \frac{4}{3} \pi R^3

Volume of a sphere of radius R.

R
R is the radius of the ball, from the centre to the surface.
Legend:
  • r — Radius (R)

Surface area of a sphere

Grade 11
S=4πR2S = 4 \pi R^2

Surface area of a sphere of radius R.

R
The area of the entire outer surface of the ball.
Legend:
  • r — Radius (R)

Diagonal of a rectangular box

Grade 11
d=a2+b2+c2d = \sqrt{a^2+b^2+c^2}

Diagonal of a rectangular box from its three dimensions.

dacb
d is the diagonal connecting opposite vertices.
Legend:
  • a — Edge a
  • b — Edge b
  • c — Edge c

Diagonal of a cube

Grade 11
d=a3d = a\sqrt{3}

Diagonal of a cube from its edge length.

daaa
The diagonal of a cube passes through its centre.
Legend:
  • a — Edge a

Total surface area of a regular prism

Grade 11
S=2S0+P0⋅hS = 2S_0 + P_0 \cdot h

Sum of the areas of the two bases and the lateral surface of a right prism.

hS
Total surface = lateral surface plus two bases.
Legend:
  • Sb — Base area (S)
  • Pb — Perimeter (P)
  • h — Height (h)

Total surface area of a cylinder

Grade 11
S=2πr(r+h)S = 2\pi r(r+h)

Sum of the lateral surface area and the two bases of a cylinder.

rh
Total surface = lateral surface plus two circles.
Legend:
  • r — Radius (r)
  • h — Height (h)

Area of the axial cross-section of a cylinder

Grade 11
S=2rhS = 2rh

Area of the rectangle formed by a cylinder's axial cross-section.

drh
The axial cross-section is a rectangle through the cylinder's axis.
Legend:
  • r — Radius (r)
  • h — Height (h)

Diagonal of the axial cross-section of a cylinder

Grade 11
D=(2r)2+h2D = \sqrt{(2r)^2+h^2}

Diagonal of the rectangle of a cylinder's axial cross-section (via the Pythagorean theorem).

drh
d is the diagonal of the cylinder's axial cross-section.
Legend:
  • r — Radius (r)
  • h — Height (h)

Surface area of a sphere via diameter

Grade 11
S=πd2S = \pi d^2

Surface area of a sphere expressed through its diameter.

Legend:
  • d — Diameter (d)

Area of a spherical cap's curved surface

Grade 11
S=2πRhS = 2\pi R h

Area of the part of a sphere's surface cut off by a plane, of height h.

Legend:
  • R — Radius (R)
  • h — Height of the segment (h)

Volume of a spherical cap

Grade 11
V=πh2(R−h3)V = \pi h^2 \left(R - \frac{h}{3}\right)

Volume of the part of a ball cut off by a plane, where h is the cap's height.

Legend:
  • R — Radius (R)
  • h — Height of the segment (h)

Volume of a spherical sector

Grade 11
V=23πR2hV = \frac{2}{3}\pi R^2 h

Volume of the solid made of a spherical cap and a cone, where h is the height of the corresponding cap.

Legend:
  • R — Radius (R)
  • h — Height of the segment (h)

Lateral surface area of a pyramid

Grade 11
Slat=12P⋅lS_{lat} = \frac{1}{2} P \cdot l

Lateral surface of a regular pyramid via the base perimeter and the apothem.

lP
l is the apothem (the height of a lateral face), P is the base perimeter.
Legend:
  • P — Base perimeter (P)
  • l — Apothem (l)

Total surface area of a pyramid

Grade 11
Sfull=Slat+SbaseS_{full} = S_{lat} + S_{base}

Total surface of a pyramid — the lateral surface plus the base area.

SбокSосн
Total surface = lateral surface plus the base.
Legend:
  • Sbok — Lateral surface
  • Sosn — Base area

Total surface area of a cone

Grade 11
S=πrl+πr2=πr(l+r)S = \pi r l + \pi r^2 = \pi r (l + r)

Total surface of a cone — the lateral surface plus the base area.

rl
Total surface = lateral surface plus the circular base.
Legend:
  • r — Radius (r)
  • l — Slant height (l)

Lateral surface area of a frustum of a cone

Grade 11
Slat=π(R1+R2)lS_{lat} = \pi (R_1 + R_2) l

Lateral surface of a frustum of a cone from the base radii and the slant height.

R₁R₂l
R₁ and R₂ are the radii of the two bases, l is the slant height.
Legend:
  • R1 — Radius R1
  • R2 — Radius R2
  • l — Slant height (l)

Total surface area of a frustum of a cone

Grade 11
Sfull=π(R1+R2)l+πR12+πR22S_{full} = \pi (R_1 + R_2) l + \pi R_1^2 + \pi R_2^2

Total surface of a frustum of a cone — the lateral surface plus the areas of the two bases.

R₁R₂l
Total surface = lateral surface plus the two bases.
Legend:
  • R1 — Radius R1
  • R2 — Radius R2
  • l — Slant height (l)

Sum of edge lengths of a regular n-gonal pyramid

Grade 11
L=n(a+b)L = n(a + b)

Sum of the lengths of all edges of a regular pyramid: n base sides plus n lateral edges.

ab
a is the base side, b is the lateral edge (n of each).
Legend:
  • n — Number of base sides (n)
  • a — Base side (a)
  • b — Lateral edge (b)

Sum of edge lengths of a regular n-gonal prism

Grade 11
L=n(2a+h)L = n(2a + h)

Sum of the lengths of all edges of a regular prism: the edges of the two bases plus the lateral edges.

ah
a is the base side, h is the height (lateral edge) of the prism.
Legend:
  • n — Number of base sides (n)
  • a — Base side (a)
  • h — Height of the prism (h)

Sum of edge lengths of a regular tetrahedron

Grade 11
L=6aL = 6a

Sum of the lengths of all 6 edges of a regular tetrahedron.

a
A tetrahedron has 4 triangular faces, all 6 edges equal a.
Legend:
  • a — Edge a

Sum of edge lengths of a regular octahedron

Grade 11
L=12aL = 12a

Sum of the lengths of all 12 edges of a regular octahedron.

a
An octahedron has 8 triangular faces, all 12 edges equal a.
Legend:
  • a — Edge a