Triangle Formulas

Everything about triangles in one place: the angle sum, exterior angle and triangle inequality, the Pythagorean theorem, the laws of sines and cosines, Heron's formula, midsegments, similarity and Thales' theorem. Many entries come with a labelled diagram and an interactive calculator.

26 formulas · Grades 5–9

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All 26 formulas on this page
  1. Perimeter of an arbitrary triangle
  2. Perimeter of an equilateral triangle
  3. Sum of the angles of a triangle
  4. Exterior angle of a triangle
  5. Triangle inequality
  6. Perimeter of an isosceles triangle
  7. Property of adjacent angles
  8. Property of vertical angles
  9. Property of an isosceles triangle
  10. Pythagorean theorem
  11. Area of a triangle
  12. Area of a right triangle
  13. Midsegment of a triangle
  14. Similarity ratio of triangles
  15. Ratio of areas of similar figures
  16. Thales' theorem (fourth proportional segment)
  17. Area of an equilateral triangle
  18. Ratio of perimeters of similar figures
  19. Triangle similarity test (SAS)
  20. Law of cosines
  21. Law of sines
  22. Area of a triangle (via an angle)
  23. Heron's formula
  24. Area of a triangle (via circumradius)
  25. Area of a triangle (via inradius)
  26. Perimeter of a triangle (via inradius and area)

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)

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