Perimeter of a rectangle

The general rule is simpler than the formula

For any shape with straight sides, the perimeter is the sum of the sides. That is the definition, and it never fails.

$$8 + 5 + 8 + 5 = 26$$

The rectangle formula is that addition grouped, nothing more:

$$\underbrace{8 + 8}_{2l} + \underbrace{5 + 5}_{2w} = 2l + 2w$$

A square has four equal sides, so $P = 4s$ for the same reason.

Units are never squared

Perimeter is a distance, so it is measured in cm or m. A perimeter given in cm² has been confused with an area, and the units alone reveal it.

Perimeter tells you almost nothing about area

The two vary independently, which surprises people.

Rectangle	Perimeter	Area
$1 \times 11$	$24$	$11$
$4 \times 8$	$24$	$32$
$6 \times 6$	$24$	$36$

Same fence, triple the field. For a fixed perimeter the area is largest when the shape is squarest — which is why fields, rooms and packaging tend towards square.

Rearranged

$$l = \frac{P - 2w}{2} \qquad w = \frac{P - 2l}{2}$$

In practice: subtract the pair you know from the perimeter, then halve what is left. A perimeter of $30$ with a length of $9$ leaves $30 - 18 = 12$ for two widths, so each is $6$.

Read the question first

“Perimeter” and “how much fencing” are usually the same number — but not when a side is left open for a gate or a wall. See perimeter.