Maths: Area and Perimeter

In these questions, you're given a shape, or its sides in words, and asked for its area or its perimeter.

The question might not say area or perimeter. It might ask how many squares fit inside, or how far it is all the way round. There are five answers to choose from.

The video above and the worked example below work through the same question, with the options in the same order. Then both work a second, shorter question: a perimeter, given in words.

The method

Four steps: inside or round, find every side, split it into rectangles, check the units.

Worked example

This shape is made from a rectangle with a corner cut away. All the sides meet at right angles. What is the area of the shape?

The drawing (not to scale): a rectangle with its top-left corner cut away. The bottom is 12 cm and the right side is 8 cm. From the bottom-left corner, the left side goes up (not labelled), then the edge steps 4 cm across to the right, then 5 cm up, then the top runs across to the right side (not labelled).

A. 20 cm² B. 40 cm² C. 76 cm² D. 96 cm² E. 116 cm²

1. Inside or round? This question asks for the area, so we want the space inside.
2. Find every side. This shape has two sides that aren't labelled. The right side is 8, so going up the left must make 8 too. The 5 is part of that, so the missing side is 8 − 5 = 3. The bottom is 12, so going along the top must make 12 too. This time the 4 is the known part, so the top is 12 − 4 = 8.
3. Split it into rectangles. Picture a line straight down from the inside corner. It makes two rectangles. 4 × 3 = 12, and 8 × 8 = 64. 12 + 64 = 76. Or start with the whole rectangle, and take the corner away. The whole rectangle is 12 by 8, which is 96. That's option D, but it still has the corner in, so D is out. The corner is 4 by 5, which is 20. That's option A, but it's only the corner, so A is out. 96 − 20 = 76 again. Add the corner on instead, and you get 116, so E is out too.
4. Check the units. Every side here is in centimetres, so the area is 76 cm². B says 40, but 40 is the distance all the way round: 12, 8, 8, 5, 4 and 3. That's the perimeter, not the space inside, so B is out. So the answer is C.

Answer: C — 76 cm² (4 × 3 + 8 × 8 = 12 + 64 = 76, or 96 − 20 = 76)

A second worked example

One more, this time a perimeter. The question is in words only; the video pictures the photo as a rectangle.

A rectangular photo is 20 cm long and 11 cm wide. What is its perimeter?

A. 220 cm B. 62 cm C. 51 cm D. 42 cm E. 31 cm

1. Inside or round? Round: the perimeter is the edge all the way round. A is 220, which is 20 × 11, the space inside, so A is out.
2. Find every side. A rectangle has four sides, two lengths and two widths, even when only two are given. 20 + 11 + 20 + 11 = 62. E is 31, just one length and one width, so E is out. C and D each leave out a side, so they're out too.
3. Split it into rectangles. Not needed here. The photo is already one rectangle, and the perimeter is round the edge, not the space inside.
4. Check the units. Round the edge is in centimetres, so it's 62 cm. So the answer is B.

Answer: B — 62 cm (20 + 11 + 20 + 11 = 62 cm)

Things to watch out for

1. Inside, or all the way round? Squares inside are area. The edges round are perimeter. A rectangle is drawn on centimetre squared paper. Its area is the squares inside: 3 rows of 5, which is 15 cm². Counting the edges all the way round gives 16. That's the perimeter, and it's one of the answers too.
2. Every side, every time. Four sides, even when only one is labelled. This square has just one side labelled, 7 cm, but all four sides are 7. All the way round, that's 7, 14, 21, 28. Stop after one side, two or three, and you land on a wrong answer.
3. Take the cut-out away. The whole rectangle, less the piece. Here the whole square is 9 by 9, which is 81, and the corner cut away is 3 by 2, which is 6. 81 still has the corner in, and 87 adds it on. Take it away: 81 − 6 = 75.
4. Same units first. Change the lengths before you multiply. A rectangle is 2.5 m long and 40 cm wide. 2.5 m is 250 cm, so the area is 250 × 40 = 10,000 cm². 2.5 × 40 is only 100, because the metres were never changed. And never change an area like a length: this rectangle is 1 m², which is 10,000 cm², not 100.

Related

• The complete 11+ question-type guide
• Maths: Shapes and Angles — name the facts, then test every option
• Maths: Measures — spot the units, then make them the same
• Maths: Coordinates and Transformations — across first, then up

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