Area Formulas for Plane Figures

Area formulas for rectangles, squares, parallelograms, rhombuses, triangles, trapezoids, regular polygons and more, often given in several equivalent forms — by sides, height, diagonals or an angle. Pick the version that matches the data in your problem.

26 formulas · Grades 5–9

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All 26 formulas on this page
  1. Area of a rectangle
  2. Area of a square
  3. Area of a parallelogram
  4. Area of a triangle
  5. Area of a right triangle
  6. Area of a rhombus (via diagonals)
  7. Area of a trapezoid
  8. Diagonals of a parallelogram
  9. Sum of angles on one side of a parallelogram
  10. Diagonal of a square
  11. Diagonal of a rectangle
  12. Area of a rhombus (via side and height)
  13. Area of a trapezoid (via midsegment and height)
  14. Area of a triangle (via an angle)
  15. Heron's formula
  16. Area of a circumscribed polygon
  17. Area of a regular polygon
  18. Area of a square (via diagonal)
  19. Area of a parallelogram (via sides and angle)
  20. Area of a parallelogram (via diagonals and angle)
  21. Area of a rhombus (via side and angle)
  22. Area of a triangle (via circumradius)
  23. Area of a triangle (via inradius)
  24. Area of a trapezoid (via diagonals and angle)
  25. Area of a regular polygon (via side)
  26. Area of an ellipse

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 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

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

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

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 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

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 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 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)

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)

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)

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