Plane Geometry

Find area and perimeter formulas for plane shapes.

Area and perimeter at a glance

Plane shapes, area and perimeter formulas, measurements, and properties
No	Shape	Area	Perimeter	Measurements and properties
1	Square	$s^2$	$4s$	$s:\text{ side length}$ Four equal sides and four right angles.
2	Rectangle	$lw$	$2(l+w)$	$l:\text{ length},\quad w:\text{ width}$ Opposite sides are equal; all angles are right angles.
3	Triangle	$\frac12 bh$	$a+b+c$	$a,b,c:\text{ sides};\quad h:\text{ height to base }b$ Height is perpendicular to the chosen base, even when it falls outside the triangle.
4	Right triangle	$\frac12 ab$	$a+b+\sqrt{a^2+b^2}$	$a,b:\text{ perpendicular legs}$ The longest side is the hypotenuse.
5	Equilateral triangle	$\frac{\sqrt3}{4}s^2$	$3s$	$s:\text{ side length}$ All three sides and all three angles are equal.
6	Parallelogram	$bh$	$2(a+b)$	$a,b:\text{ adjacent sides};\quad h:\text{ height to }b$ Use perpendicular height, not the slanted side, for area.
7	Rhombus	$\frac12 d_1d_2$	$4s$	$d_1,d_2:\text{ diagonals};\quad s:\text{ side}$ All sides are equal; the diagonals meet at right angles.
8	Kite	$\frac12 d_1d_2$	$2(a+b)$	$d_1,d_2:\text{ diagonals};\quad a,b:\text{ distinct side lengths}$ A convex kite has two pairs of equal adjacent sides and perpendicular diagonals.
9	Trapezoid (trapezium)	$\frac12(a+b)h$	$a+b+c+d$	$a,b:\text{ parallel bases};\quad c,d:\text{ legs};\quad h:\text{ height}$ Height is the perpendicular distance between the parallel bases.
10	Circle	$\pi r^2$	$2\pi r$	$r:\text{ radius}$ The perimeter of a circle is called its circumference.
11	Semicircle	$\frac12\pi r^2$	$\pi r+2r$	$r:\text{ radius}$ Perimeter includes the curved arc and the straight diameter.
12	Regular polygon	$\frac12 nst$	$ns$	$n:\text{ sides};\quad s:\text{ side length};\quad t:\text{ apothem}$ All sides and angles are equal. The apothem runs from the center perpendicular to a side; the sketch is a hexagon.

Use consistent measurements

Convert all lengths to the same unit before substituting. Perimeter uses length units such as centimeters; area uses square units such as $\mathrm{cm}^2$ . All lengths are positive, and the polygon formulas assume simple, nondegenerate shapes.

For a rectangle with $l=8\,\mathrm{cm}$ and $w=3\,\mathrm{cm}$ :

A=8\cdot3=24\,\mathrm{cm}^2,\qquad P=2(8+3)=22\,\mathrm{cm}

Keep $\pi$ and square roots exact until the final step. For a composite shape, add nonoverlapping areas or subtract cutouts. Count only the exposed boundary when finding its perimeter.