Perimeter, Area, and Volume

1. The perimeter of a polygon (or any other closed curve, such as a circle) is the distance around the outside.

2. The area of a simple, closed, planar curve is the amount of space inside.

3. The volume of a solid $3 D$ shape is the amount of space displaced by it.

Some formulas for common $2$ -dimensional plane figures and $3$ -dimensional solids are given below. The answers have one, two, or three dimensions; perimeter is measured in linear units , area is measured in square units , and volume is measured in cubic units .

Table $1$ . Perimeter Formulas
Shape	Formula	Variables
Square	$P = 4 s$	$s$ is the length of the side of the square.
Rectangle	$P = 2 L + 2 W$	$L$ and $W$ are the lengths of the rectangle's sides (length and width).
Triangle	$a + b + c$	$a, b$ , and $c$ are the side lengths.
Right Triangle, with legs $a$ and $b$ (see Pythagorean Theorem )	$P = a + b + \sqrt{a^{2} + b^{2}}$	$a$ and $b$ are the lengths of the two legs of the triangle
Circle	$P = C = 2 π r = π d$	$r$ is the radius and $d$ is the diameter.

Table 2. Area Formulas
Shape	Formula	Variables
Square	$A = s^{2}$	$s$ is the length of the side of the square.
Rectangle	$A = L W$	$L$ and $W$ are the lengths of the rectangle's sides (length and width).
Triangle	$A = \frac{1}{2} b h$	$b$ and $h$ are the base and height
Triangle	$\begin{array}{l} A = \sqrt{s (s - a) (s - b) (s - c)} \\ where s = \frac{a + b + c}{2} \end{array}$	$a$ , $b$ , and $c$ are the side lengths and $s$ is the semiperimeter
Parallelogram	$A = b h$	$b$ is the length of the base and $h$ is the height.
Trapezoid	$A = \frac{b_{1} + b_{2}}{2} h$	$b_{1}$ and $b_{2}$ are the lengths of the parallel sides and $h$ the distance (height) between the parallels.
Circle	$A = π r^{2}$	$r$ is the radius.

Table 3. Volume Formulas
Shape	Formula	Variables
Cube	$V = s^{3}$	$s$ is the length of the side.
Right Rectangular Prism	$V = L W H$	$L$ is the length, $W$ is the width and $H$ is the height.
Prism or Cylinder	$V = A h$	$A$ is the area of the base, $h$ is the height.
Pyramid or Cone	$V = \frac{1}{3} A h$	$A$ is the area of the base, $h$ is the height.
Sphere	$V = \frac{4}{3} π r^{3}$	$r$ is the radius.

Perimeter, Area, and Volume

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