# 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 $3\text{D}$ 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 $P=4s$ $s$ is the length of the side of the square. Rectangle $P=2L+2W$ $L$ 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 $P=C=2\pi r=\pi d$ $r$ 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 $A=LW$ $L$ and $W$ are the lengths of the rectangle's sides (length and width). Triangle $A=\frac{1}{2}bh$ $b$ and $h$ are the base and height Triangle $\begin{array}{l}A=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}\\ \text{where}\text{\hspace{0.17em}}\text{\hspace{0.17em}}s=\frac{a\text{\hspace{0.17em}}+\text{\hspace{0.17em}}b\text{\hspace{0.17em}}+\text{\hspace{0.17em}}c}{2}\end{array}$ $a$ , $b$ , and $c$ are the side lengths and $s$ is the semiperimeter Parallelogram $A=bh$ $b$ is the length of the base and $h$ is the height. Trapezoid $A=\frac{{b}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{b}_{2}}{2}h$ ${b}_{1}$ and ${b}_{2}$ are the lengths of the parallel sides and $h$ the distance (height) between the parallels. Circle $A=\pi {r}^{2}$ $r$ is the radius.

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