These questions ask you to measure the surface inside a shape built from rectangles or left after a rectangular piece is removed. You may also need to compare an area method with a boundary-length method.
Choose a clean rectangle method
First mark what is present and what is absent. If the shape is already described as separate rectangles that touch without overlapping, multiply the two side lengths of each rectangle and add the results. For example, rectangles measuring 7 by 3 and 2 by 4 cover 21 plus 8, or 29 units of area. A shared edge has no area, so it is not added or removed.
If the shape is a large rectangle with a rectangular notch or hole, subtraction is often shorter. An 11 by 6 rectangle with a 3 by 2 piece missing has area 66 minus 6, or 60 units of area. Splitting the remaining shape is equally sound: every included piece is counted once, and the sum must match the subtraction route.
| Signal in the description | Useful move |
|---|---|
| Non-overlapping rectangles are joined | Find every rectangular area, then add. |
| A rectangle is cut from a larger rectangle | Find the whole area and subtract the cut area. |
| A side of a smaller rectangle is not stated | Subtract the known part from the matching total length. A total of 13 with a notch of 5 leaves 8. |
| The question asks how much surface is covered | Use area units such as cm² or m². If it asks for the distance around the edge, add boundary lengths instead. |
Avoid these traps
| Tempting route | How to reject it |
|---|---|
| Stop after finding the largest rectangular piece | Check off every region that belongs to the shape. Any smaller joined piece still contributes its own area. |
| Treat a row of equal tiles as one much larger tile | Joining tiles in one direction changes the total length but not the height. Add the small tile areas instead. |
| Use the enclosing rectangle for a stepped shape | An enclosing rectangle may contain empty regions. Count it only after subtracting every missing region. |
| Subtract the missing side lengths | A cut removes a surface, not two lengths. Multiply the cut dimensions before subtracting. |
| Keep the complete rectangle after a piece is removed | The whole is only a starting value. The removed area must leave the total. |
| Use an area value for a ribbon around the edge | Area uses units such as cm² or m²; a route around the boundary uses ordinary length units. |
Try it
A garden path is made of two rectangular slabs laid end to end in a straight line. One slab measures 10 metres by 3 metres and the other measures 5 metres by 3 metres. What area does the path cover?
- 15 square metres
- 30 square metres
- 36 metres
- 45 square metres
A rectangular sheet of paper measures 12 centimetres by 8 centimetres. A rectangular hole 4 centimetres by 3 centimetres is cut out of the middle of it. Enter the number of square centimetres of paper that remain.
A sheet of card 14 centimetres across and 9 centimetres tall has a piece 5 centimetres across and 4 centimetres tall missing from its top right corner. A learner wants to split the remaining L-shape into two rectangles rather than subtract the missing piece. Select all the statements that are true about this splitting method.
- The right-hand piece is 5 centimetres across and 9 centimetres tall, since the sheet is 9 centimetres tall.
- A vertical cut 9 centimetres from the left edge leaves a 9 by 9 rectangle and a 5 by 5 rectangle.
- Splitting a shape into rectangles gives a different total from subtracting, so only one of the two methods can be trusted.
- Adding 81 and 25 gives the same total as taking the missing corner away from the whole sheet.