Perimeter

How much fencing for a flower bed? How much trim for a picture frame? How far is one lap of the track? Every one of these questions has the same answer: the perimeter — the distance you cover walking once around the edge.

Add up the whole trip

Finding a perimeter needs no new tools at all: add the sides. A vegetable plot shaped like a trapezoid has sides 3 m, 5 m, 4 m and 6 m, so one lap measures

$3 + 5 + 4 + 6 = 18 \text{ m}$

Count the sides, measure them, add. That is all a perimeter ever asks, of any shape.

Let a formula add for you

A rectangle's opposite sides are equal, so its four sides come in two matching pairs:

$P = 2(w + h)$

A rectangle 6 cm wide and 4 cm tall has $P = 2 \times (6 + 4) = 20$ cm. A square's four sides all match, so $P = 4s$ . Regular polygons are even friendlier: a regular hexagon with side 5 cm has perimeter $6 \times 5 = 30$ cm — number of sides times side length, $P = n \times s$ .

Perimeter is not area

Perimeter measures the trip around; area measures the covering inside (a story told at Area), and even the units differ: cm versus cm². The most striking proof: equal perimeters can enclose very different areas.

Rectangles with perimeter 16 cm come in many shapes:

$2 \times 6$ : perimeter 16, area 12;
$3 \times 5$ : perimeter 16, area 15;
$4 \times 4$ : perimeter 16, area 16 — the largest.

The same fence catches the most ground when the shape is as square as possible.

InteractivePerimeter & Area

Width w5Height h4

Area = length × width=5 × 4 = 20unit squares

Arrange rectangles and triangles on the grid: the cells count the area, while one lap around the edge traces the perimeter — one shape, two ways of measuring. Drag width and height and watch the two numbers follow different commands.

Old farmer's wisdom

For a fixed perimeter, the closer the shape is to a circle, the more area it encloses; the square comes next, and long thin strips do worst. That is why beds are laid out round or square, almost never as narrow ribbons.

Perimeter in daily life

Fences and rails: one lap around the plot tells you how much material to buy;
Frames and trim: the border of a picture is exactly its perimeter;
One lap of the track: hugging the innermost line, that lap is the track's perimeter.

Circles have a perimeter too, with a formula of their own, $C = 2\pi r$ — how it connects to π is a story told at Circles.

Check yourself

Quick quiz

1. A rectangle is 7 by 3. What is its perimeter?
2. A regular pentagon has side 4. What is its perimeter?
3. Two rectangles both have perimeter 16: one is 2×6, the other 4×4. Which has the larger area?