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.
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. 
$P=a+b+\sqrt{{a}^{2}+{b}^{2}}$ 
$a$ and $b$ are the lengths of the two legs of the triangle  
Circle 

$r$ is the radius and $d$ is the diameter. 
Shape 
Formula 
Variables 
Square 

$s$ is the length of the side of the square. 
Rectangle 

$L$ 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 
$\begin{array}{l}A=\sqrt{s\left(\mathrm{s\; \; a}\right)\left(\mathrm{s\; \; b}\right)\left(\mathrm{s\; \; c}\right)}\\ \text{where}s=\frac{a+b+c}{2}\end{array}$ 
$a$, $b$, and $c$ are the side lengths and $s$ is the semiperimeter 
Parallelogram 

$b$ 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 
$A=\pi {r}^{2}$ 
$r$ is the radius. 
Shape 
Formula 
Variables 
Cube 

$s$ is the length of the side. 
Right Rectangular Prism 

$L$ 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 

$r$ is the radius. 