Distance around
Perimeter is the distance all the way around the outside of a shape. Imagine walking along the edge of a field and back to where you started: the distance you walked is the perimeter.
Perimeter is a length, so it’s measured in cm, m, km and so on.
Finding perimeter: add all the sides
For any polygon, add up every side length.
Example: A triangle with sides 5 cm, 7 cm and 9 cm.
Perimeter = 5 + 7 + 9 = 21 cm.
Rectangles
A rectangle has two pairs of equal sides: length (l) and width (w).
Perimeter = l + w + l + w = 2l + 2w = 2(l + w)
Example: 8 m by 3 m.
P = 2(8 + 3) = 2 × 11 = 22 m.
Squares and regular polygons
A square has 4 equal sides: P = 4s.
A regular polygon with n equal sides: P = n × s.
| Shape | Side | Perimeter |
|---|---|---|
| Square | 6 cm | 4 × 6 = 24 cm |
| Regular pentagon | 3 m | 5 × 3 = 15 m |
| Regular hexagon | 10 cm | 6 × 10 = 60 cm |
These are formulas: rules that work every time.
Missing sides
If you know the perimeter, you can find a missing side.
Example: A rectangle has perimeter 30 cm and length 9 cm. Find the width.
- 2 lengths = 18 cm.
- 30 − 18 = 12 cm for the 2 widths.
- Width = 6 cm.
Shapes on a grid
For a shape drawn on grid paper, count the unit edges around the outside. Don’t count the squares inside: that’s area.
L-shapes and other composite shapes
Find any missing side lengths first, then add every outside edge.
Example: Start with a rectangle 10 m wide and 8 m tall, and cut a 6 m by 5 m notch out of its top-right corner. Going around the L-shape, the sides are:
| Side | Length |
|---|---|
| Bottom | 10 m |
| Right (short) | 8 − 5 = 3 m |
| Step across | 6 m |
| Step up | 5 m |
| Top | 10 − 6 = 4 m |
| Left | 8 m |
Perimeter = 10 + 3 + 6 + 5 + 4 + 8 = 36 m. Notice that this equals the perimeter of the original 10 by 8 rectangle: cutting a notch from a corner moves edges inward without changing the total.
Real life
- Fencing a garden
- Putting a frame around a picture
- Running laps around a track
- Trim around a room
Worked examples
Example 1: Find the perimeter of a rectangle 12 cm by 5 cm.
2(12 + 5) = 34 cm.
Example 2: A square has perimeter 36 m. How long is each side?
36 ÷ 4 = 9 m.
Example 3: A garden is 7 m by 4 m. Fencing costs $10 per metre. How much?
P = 2(7 + 4) = 22 m. 22 × $10 = $220.
Common mistakes
- Adding only two sides of a rectangle (length + width) and forgetting the other two.
- Mixing up perimeter and area. Perimeter is around; area is inside.
- Forgetting units, or mixing cm and m.
Try it yourself
- Perimeter of a triangle with sides 4 cm, 4 cm and 6 cm.
- Perimeter of a square with side 15 m.
- A rectangle is 20 cm long with perimeter 56 cm. Find its width.
- Perimeter of a regular octagon with side 3 cm.
Answers: 1) 14 cm 2) 60 m 3) 8 cm 4) 24 cm
Quick recap
- Perimeter is the distance around a shape: add all side lengths.
- Rectangle: P = 2(l + w). Square: P = 4s.
- Find missing sides before adding.