ShapesArea and perimeter: what each one measures
Area and perimeter are both measures of a flat shape, and they are mixed up more than any other pair in school maths. They measure different things, in different units, and one does not decide the other. This page sorts them out.
Perimeter is the edge
The perimeter is the total length of the boundary: the distance you would walk going once round the shape. It is a length, so it is measured in centimetres, metres or kilometres. Fencing a field, putting a border on a sari, framing a picture and running laps of a ground are perimeter jobs.
Area is the surface inside
The area is the amount of flat surface the shape covers. It is measured in square units: the number of unit squares needed to cover the shape. Painting a wall, tiling a floor, buying cloth for a tablecloth and sowing a field are area jobs.
| Perimeter | Area | |
|---|---|---|
| Measures | the boundary | the surface inside |
| Units | cm, m, km | cm², m², km², hectares |
| Rectangle | 2 × (length + breadth) | length × breadth |
| Square | 4 × side | side × side |
| Circle | 2πr (circumference) | πr² |
| Everyday job | fencing, edging, framing | painting, tiling, carpeting |
Same perimeter, different areas
Take 24 m of fencing. You can enclose many rectangles with it.
| Length | Breadth | Perimeter | Area |
|---|---|---|---|
| 11 m | 1 m | 24 m | 11 m² |
| 10 m | 2 m | 24 m | 20 m² |
| 9 m | 3 m | 24 m | 27 m² |
| 8 m | 4 m | 24 m | 32 m² |
| 7 m | 5 m | 24 m | 35 m² |
| 6 m | 6 m | 24 m | 36 m² |
The perimeter never changes, but the area runs from 11 m² to 36 m². The square encloses the most. That is a general rule: among all rectangles with the same perimeter, the square has the greatest area. Among all shapes of any kind, the circle does.
Same area, different perimeters
Now fix the area at 36 m².
| Length | Breadth | Area | Perimeter |
|---|---|---|---|
| 36 m | 1 m | 36 m² | 74 m |
| 18 m | 2 m | 36 m² | 40 m |
| 12 m | 3 m | 36 m² | 30 m |
| 9 m | 4 m | 36 m² | 26 m |
| 6 m | 6 m | 36 m² | 24 m |
A long thin plot needs far more fencing than a square one of the same area.
What happens when a shape is enlarged
Double every length of a shape and its perimeter doubles, but its area becomes four times as large. Triple the lengths, and the area becomes nine times as large. A photograph printed at twice the width and height uses four times the paper.
Shapes made of rectangles
For an L-shaped room, cut the shape into rectangles, find each area and add. For the perimeter, do not add the perimeters of the pieces, because the cut lines are inside the shape. Walk round the outside edge only, adding each side once.
Units to watch
Lengths must be in the same unit before you calculate. And remember that 1 m² is not 100 cm² but 10,000 cm², because a square metre is 100 cm long and 100 cm wide. More on the units page.
Open any shape from the home page for its formulas, or use the calculator. For practice sheets, see area and perimeter worksheets.