Perimeter: Meaning, Formulas for Common Shapes, and One Example

Perimeter is the total distance around the outside of a two-dimensional shape. For polygons, you find it by adding the side lengths. For a circle, the perimeter is called the circumference.

If you want the short version, it is this:

$$P = \text{sum of the outside side lengths}$$

For a circle, use:

$$C = 2\pi r = \pi d$$

where $r$ is the radius and $d$ is the diameter.

What Perimeter Measures

Perimeter measures boundary length, not enclosed space. That is why it is different from area.

If you are asking how much fencing, trim, border, or edging is needed, perimeter is usually the right measurement. If you are asking how much surface is covered, you want area instead.

Perimeter Formulas for Common Shapes

These formulas are just shortcuts for adding the outside lengths.

Square

If each side has length $s$, then:

$$P = 4s$$

Rectangle

If the length is $l$ and the width is $w$, then:

$$P = 2l + 2w = 2(l+w)$$

Triangle

If the side lengths are $a$, $b$, and $c$, then:

$$P = a+b+c$$

Regular Polygon

If a regular polygon has $n$ equal sides of length $s$, then:

$$P = ns$$

This works because every side has the same length. If the polygon is not regular, add the side lengths one by one instead.

Circle

The perimeter of a circle is its circumference:

$$C = 2\pi r$$

or

$$C = \pi d$$

Both formulas mean the same thing because $d=2r$.

Worked Example: Rectangle Perimeter

Suppose a garden is $9$ meters long and $4$ meters wide. To find the fencing needed around it, use the rectangle formula:

$$P = 2(l+w)$$

Substitute $l=9$ and $w=4$:

$$P = 2(9+4) = 2(13) = 26$$

So the perimeter is $26$ meters. The unit stays in meters, not square meters, because perimeter is a length.

This example shows the main idea clearly: perimeter means going all the way around the boundary once.

Common Mistakes with Perimeter

• Mixing up perimeter and area. Perimeter is measured in units such as centimeters or meters, while area is measured in square units.
• Forgetting to use the whole boundary. In a rectangle, you need both pairs of sides.
• Mixing units before adding. Convert first if one side is in centimeters and another is in meters.
• Using $2\pi r$ for shapes that are not circles.
• Assuming $ns$ works for any polygon. It only works directly when all $n$ sides have the same length.

When to Use Perimeter Instead of Area

Use perimeter when the edge length matters more than the inside region. Common examples include fencing a yard, adding trim around a room, putting a border around a poster, or finding the distance around a track.

It also appears in later math courses. In coordinate geometry, for example, you may first find side lengths with the distance formula and then add them to get the perimeter.

Try a Similar Perimeter Problem

Try your own version with a triangle whose side lengths are $5$, $7$, and $9$. Add the three sides and check that your answer is in linear units.

If you want to explore another case with your own numbers, solve a similar perimeter problem in GPAI Solver.