Area and Perimeter Formulas Reference

Rectangle and square

\text{Rectangle: Area} = l \times b, \qquad \text{Perimeter} = 2(l+b)

\text{Square: Area} = a^2, \qquad \text{Perimeter} = 4a

Circle and sector

\text{Circle: Area} = \pi r^2, \qquad \text{Circumference} = 2\pi r

A sector is the pie-slice region enclosed by two radii and an arc, spanning an angle $C$ in degrees:

\text{Area of sector} = \pi r^2 \times \dfrac{C}{360}

\text{Perimeter of sector} = 2r + 2\pi r \times \dfrac{C}{360}

Triangle, including Heron's formula

With a known base and height, area is straightforward. When only the three side lengths are known, Heron's formula avoids needing the height at all:

\text{Area} = \dfrac{1}{2} \times \text{base} \times \text{height}

s = \dfrac{a+b+c}{2}, \qquad \text{Area} = \sqrt{s(s-a)(s-b)(s-c)}

Worked example: find the area of a triangle with sides 5, 6, and 7.

s = \dfrac{5+6+7}{2} = 9

\text{Area} = \sqrt{9(9-5)(9-6)(9-7)} = \sqrt{9 \times 4 \times 3 \times 2} = \sqrt{216} = 6\sqrt{6}

Regular pentagon

A regular pentagon splits neatly into 5 identical triangles, so its area is 5 times one of them:

\text{Area} = 5 \times \left(\dfrac{1}{2} \times \text{base} \times \text{height}\right), \qquad \text{Perimeter} = 5 \times \text{side}

When only the side length $a$ is known:

\text{Area} = \dfrac{5a^2}{4\tan 36^\circ}