Shape Areas

Triangle area using height and base

$$S = \frac{1}{2}\cdot a\cdot h$$

a - side (base)
h - triangle height

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Triangle area using two sides and the angle between them

$$S = \frac{1}{2}\cdot a\cdot b\cdot sin(C)$$

a, b - sides
C - angle between a and b

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Tringle area using three sides (Heron's formula)

$$S = \sqrt {p\cdot (p-a)\cdot (p-b)\cdot (p-c)}$$

a, b, c - sides
p - half perimeter (semiperimeter)

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Triangle area using semiperimeter and inradius

$$S = p\cdot r$$

p - half perimeter (semiperimeter)
r - inradius

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Triangle area using three sides and circumradius

$$S = \frac{a\cdot b\cdot c}{4\cdot R}$$

a, b, c - sides
R - circumradius

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Right triangle area

$$S = \frac{1}{2}\cdot a\cdot b$$

a, b - legs (sides)

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Equilateral triangle area

$$S = \frac{a^{2}\cdot \sqrt {3}}{4}$$

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Area of a square

$$S = a^{2}$$

a - side

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Area of a square using diagonal

$$S = \frac{1}{2}\cdot d^{2}$$

d - diagonal

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Area of a rectangle

$$S = a\cdot b$$

a,b - sides

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Area of a rectangle using diagonal

$$S = \frac{1}{2}\cdot d^{2}\cdot sin(\phi)$$

d - diagonal
φ - angle between diagonals

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Area of a parallelogram

$$S = a\cdot h$$

a - base
h - height

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Area of a parallelogram using two sides

$$S = a\cdot b\cdot sin(\alpha)$$

a, b - sides
α - angle between a and b

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Area of a parallelogram using diagonals

$$S = \frac{1}{2}\cdot d1\cdot d2\cdot sin(\phi)$$

d1, d2 - diagonals
φ - angle between diagonals

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Area of a rhombus

$$S = a^{2}\cdot sin(\alpha)$$

a - side
α - angle between sides

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Area of a rhombus using diagonals

$$S = \frac{d1\cdot d2}{2}$$

d1, d2 - diagonals

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Area of a cyclic quadrilateral

$$S = \sqrt {(p-a)\cdot (p-b)\cdot (p-c)\cdot (p-d)}$$

a, b, c, d - sides
p - half perimeter (semiperimeter)

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Area of a trapezium (trapezoid)

$$S = \frac{(a+b)\cdot h}{2}$$

a, b - bases of a trapezoid
h - trapezoid height

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Area of a trapezium (trapezoid) using midline

$$S = m\cdot h$$

m - trapezoid midline
h - trapezoid height

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Area of a regular polygon inside circumcircle

$$S = \frac{1}{2}\cdot R^{2}\cdot n\cdot sin(\frac{360}{n})$$

R - radius of circumcircle (long radius)
n - number of angles or sides

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Area of a regular polygon

$$S = \frac{n\cdot a\cdot r}{2}$$

r - inradius
a - side
n - knumber of angles or sides

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Area of an equilateral triangle using circumradius

$$S = \frac{3\cdot \sqrt {3}\cdot R^{2}}{4}$$

R - circumradius

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Area of an equilateral triangle using inradius

$$S = 3\cdot \sqrt {3}\cdot r^{2}$$

r - inradius

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Area of an equilateral triangle using height

$$S = \frac{\sqrt {3}\cdot h^{2}}{3}$$

h - height of triangle

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Area of a square using circumradius

$$S = 2\cdot R^{2}$$

R - circumradius

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Area of a square using inradius

$$S = 4\cdot r^{2}$$

r - inradius

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Area of a regular hexagon

$$S = \frac{3\cdot \sqrt {3}\cdot a^{2}}{2}$$

a - side of a hexagon

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Area of a regular hexagon using circumradius

$$S = \frac{3\cdot \sqrt {3}\cdot R^{2}}{2}$$

R - circumradius

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Area of a regular hexagon using inradius

$$S = 2\cdot \sqrt {3}\cdot r^{2}$$

r - inradius

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Area of a circle

$$S = \pi\cdot R^{2}$$

R - radius

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Area of sector of a circle

$$S = \frac{R\cdot l}{2}$$

R - radius
l - arc length

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Area of sector of a circle using angle

$$S = \frac{\pi\cdot R^{2}\cdot \alpha}{360}$$

R - radius
α - central angle

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Area of a smaller segment of a circle

$$S = \frac{\pi\cdot R^{2}\cdot \alpha}{360}-S_{\DeltaAOB}$$

R - radius
α - central angle

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Area of a bigger segment of a circle

$$S = \frac{\pi\cdot R^{2}\cdot \alpha}{360}+S_{\DeltaAOB}$$

R - radius
α - central angle

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Area of a segment of a circle

$$S = \frac{R^{2}}{2}\cdot (\frac{\pi\cdot \alpha}{180}-sin(\alpha))$$

R - radius
α - central angle

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