# Area Formulas

The standard notation for area is A.

#### Square

The area of a square of side a is:

A = a ⋅ a = a2

#### Rectangle

If a is the length and b is the width of a rectangle its area is:

A = a ⋅ b

#### Parallelogram

Let a pair of adjacent sides of the palellogram be a and b and altitudes ha and hb.
The parallelogram area is given by the formula:

A = a ⋅ ha = b ⋅ hb

#### Trapezoid

Let the lengths of the both parallel sides of a trapezoid be a and b and the distance between them is h(the trapezoid altitude).
The area is given by the formula:

$A = \frac{1}{2}(a + b)h$

#### Area of a circle

$A = \pi\cdot r^2$

#### Area of a right triangle

$A=\frac{a\cdot b}{2}$

$A=\frac{c\cdot h_c}{2}$

#### Area of a triangle

Let ABC be a triangle

with sides of length a, b, c and
altitudes ha, hb and hc.

$A = \frac{a\cdot h_a}{2} = \frac{b\cdot h_b}{2} = \frac{c\cdot h_c}{2}$

#### Area of a triangle by 3 sides

$A = \sqrt{p(p - a)(p - b)(p - c)}$, where $p = \frac{a + b + c}{2}$

The formula is known as Heron's formula, and $p$ is called semiperimeter.

If we exclude the semiperimeter ($p$) the formula looks like:
$A=\frac14\sqrt{4a^2b^2-(a^2+b^2-c^2)^2}$

#### Area calculator

Enter the triangle:
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$$c=$$

#### Other formulas for area of a triangle

$A = \frac{a\cdot b\cdot \sin C}{2} = \frac{a\cdot c\cdot \sin B}{2} = \frac{b\cdot c\cdot \sin A}{2}$

$A = R^2\sin(A) \cdot \sin(B) \cdot \sin(C) = \frac{abc}{4R}$
where R is the radius of the circumscribed circle.

#### Area of parallelogram, rhombus

$A = AB\cdot DE = BC \cdot DF$
$A = AB \cdot AD \sin \alpha$
$A = \frac12 AC \cdot BD \sin \gamma$

$A = \frac12 AC \cdot BD \sin \varphi$

#### Area of a regular polygon

$A = \frac14 n\cdot a^2 cot(\frac{\pi}{n})$

n is the number of edges(vertices).
$\pi=3.14159265359$

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