Area and perimeter of rectangles
The perimeter is the distance around; the area is the space inside. Find both for rectangles and squares, and work backwards to a side.
If you want to put a fence around a garden, you need to know the distance around it. If you want to cover a floor with carpet, you need to know how much space it covers. These are two different questions, and they have two different answers: the perimeter and the area. Rectangles and squares are the shapes of rooms, doors, screens, books and fields, so these two ideas are among the most useful in all of maths. For rectangles, both come from just two numbers: the width and the height.
Perimeter and area are different
- The perimeter is the total length of the edge, all the way round. You find it by adding the sides.
- The area is the amount of flat space inside. It is the number of small squares of side 1 that fit inside the shape. You find it by multiplying.
The same rectangle has an area of 12 and a perimeter of 14. The two numbers measure different things, so they are usually different numbers. Even when they happen to be equal, as in a square with side 4 (area 16, perimeter 16), they still mean different things.
The area of a rectangle
Picture the rectangle filled with small squares, in rows. The width tells you how many squares are in one row. The height tells you how many rows there are. To count all the squares, multiply.
Remember
Area of a rectangle = width × height
Example 1
What is the area of a rectangle 12 wide and 5 high?
- Area = width × height.
- 12 × 5 = 60.
Answer: 60
The perimeter of a rectangle
A rectangle has four sides. The two opposite sides are always the same length. A drawing usually labels only two sides, but the other two are still there.
Remember
Perimeter of a rectangle = width + height + width + height
This is the same as 2 × (width + height): add the two different sides, then double.
Example 2
What is the perimeter of a rectangle 11 wide and 4 high?
- Add all four sides: 11 + 4 + 11 + 4 = 30.
- Or add the two labelled sides and double: 11 + 4 = 15, and 2 × 15 = 30.
Answer: 30
Squares
A square is a rectangle with four equal sides. So you only need one number, the length of a side.
Remember
Area of a square = side × side. Perimeter of a square = 4 × side.
Multiplying a number by itself is called squaring it, and it is written with a small 2: 7² = 7 × 7 = 49. That is where the name comes from: the area of a square with side 7 is "7 squared".
Example 3
A square has sides of 11. What are its area and its perimeter?
- Area: 11 × 11 = 121.
- Perimeter: the four sides are all 11, so 4 × 11 = 44.
Answer: The area is 121 and the perimeter is 44.
Working backwards: finding a side
Sometimes you know the area or the perimeter, and you need a side. Then you undo what the rule does. Multiplying is undone by dividing, and squaring is undone by finding the square root.
| You know | You want | What to do |
|---|---|---|
| The area of a square | Its side | Find the number that, multiplied by itself, gives the area. |
| The perimeter of a square | Its side | Divide the perimeter by 4. |
| The area of a rectangle and one side | The other side | Divide the area by the side you know. |
Example 4
A square has an area of 49. How long is each side?
- Which number times itself gives 49?
- 7 × 7 = 49.
- So each side is √49 = 7.
Answer: 7
Example 5
A square has a perimeter of 44. How long is each side?
- The perimeter is made of 4 equal sides.
- 44 ÷ 4 = 11.
- Check: 4 × 11 = 44.
Answer: 11
Example 6
A rectangle has an area of 56. Its width is 8. What is its height?
- Area = width × height, so 8 × ? = 56.
- Divide the area by the width: 56 ÷ 8 = 7.
- Check: 8 × 7 = 56.
Answer: 7
To find square roots quickly, it helps to know the first square numbers by heart:
| Side | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Area | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 |
Try it yourself
1.What is the area of a rectangle 13 wide and 4 high?
Show the answer
52
Area = width × height, and 13 × 4 = 52.
2.What is the perimeter of a rectangle 12 wide and 7 high?
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38
12 + 7 + 12 + 7 = 38. Or 12 + 7 = 19, and 2 × 19 = 38.
3.A square has sides of 15. What are its area and its perimeter?
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Area 225, perimeter 60
Area: 15 × 15 = 225. Perimeter: 4 × 15 = 60.
4.A square has a perimeter of 52. How long is each side?
Show the answer
13
The four sides are equal, so 52 ÷ 4 = 13.
5.A square has an area of 144. How long is each side?
Show the answer
12
12 × 12 = 144, so the side is √144 = 12.
6.A rectangle has an area of 72. One side is 9. How long is the other side?
Show the answer
8
Divide the area by the side you know: 72 ÷ 9 = 8. Check: 9 × 8 = 72.
Ready to test yourself?
Play Geometry: 20 seconds per question, and it gets harder as you go.