Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Unit 7, lesson 3 (area of triangles and parallelograms): a triangle has base 14 cm and height 5 cm. What is its area?
Answer
35 cm². ½ × 14 × 5 = 35. - From Unit 7, lesson 1 (perimeter and area of rectangles): a rectangle is 10 m by 3 m. What is its perimeter?
Answer
26 m. 10 + 3 + 10 + 3 = 26. - From Unit 1, lesson 6 (order of operations): what is 8 × 6 + 5 × 2?
Answer
58. Do both multiplications first: 48 + 10 = 58.
Words used in this lesson
- compound shape
- a shape made up of two or more simpler shapes joined together. For example: An L-shape is two rectangles.
- notch
- a piece cut out of the edge or corner of a shape. For example: A corner taken out of a rectangle.
- outside edge
- a side that is on the boundary of the shape. For example: The fence round a field.
- split
- to divide a shape into pieces with straight lines. For example: Cutting an L-shape into two rectangles.
- subtract
- to take one number away from another. For example: The whole area minus the missing piece.
- overall
- the total length or width of the whole shape. For example: An overall length of 12 m.
- trapezium
- a four-sided shape with one pair of parallel sides. For example: A paving slab that is wider at one end.
A shape with a bite out of it
A tiler in Marrakech is quoting for a courtyard that is not a rectangle: a rectangle with a corner taken out where a stairwell sits. The tiles are priced by the square metre, the edging strip by the metre. Two different numbers again, and neither comes from a ready-made formula. They come from splitting the shape into pieces you can already measure.
Predict first: if you split the notched shape two different ways, will you get two different areas? Try all three buttons.
Cut across the notch: a lower rectangle 10 m by 5 m is 50 m² and an upper rectangle 6 m by 3 m is 18 m². Together 50 + 18 = 68 m².
Show me the steps
- Notch 4 m by 3 m, cut across. The lower rectangle is 10 m by 5 m, which is 50 m². The upper rectangle is 6 m by 3 m, which is 18 m². Together 50 + 18 = 68 m².
- Press Cut down. The left rectangle is 6 m by 8 m, which is 48 m², and the right rectangle is 4 m by 5 m, which is 20 m². Together 48 + 20 = 68 m². The same area.
- Press Whole minus notch. The whole rectangle is 10 × 8 = 80 m² and the notch is 4 × 3 = 12 m². 80 − 12 = 68 m². The same again.
- Make the notch 6 m wide and 5 m high and cut across. The lower rectangle is 10 × 3 = 30 m², the upper is 4 × 5 = 20 m², and 30 + 20 = 50 m². The whole minus notch gives 80 − 30 = 50 m².
The area of a compound shape is the sum of its parts, or the whole minus what is missing.
- Split it into rectangles and triangles, find each area, and add. Any split that covers the shape exactly once gives the same total.
- Subtract: if the shape is a rectangle with a piece missing, take the missing piece's area from the whole rectangle.
- Trapezium: a four-sided shape with one pair of parallel sides, a and b, a height h apart. Split it into a rectangle and one or two triangles, or turn a copy upside down beside it: the two make a parallelogram with base (a + b) and height h, so each trapezium is ½ × (a + b) × h.
- For the perimeter, only the outside edges count. A line you drew to split the shape is inside, so it is not part of the perimeter.
Back to the courtyard: the tiles cover the area (68 m² for the notched shape in the figure), and the edging strip runs along the outside edges only. Unlike area, the perimeter changes with where the missing piece is, as the second figure shows.
Which edges count?
The edging runs along the outside only. Move the notch between a corner and the middle of an edge and watch the outside edges.
The notch is in a corner, 4 m by 3 m. The outside edges still add to 36 m, the same as the full 10 m by 8 m rectangle. The area is 68 m².
Show me the steps
- Notch 4 m by 3 m in a corner. The outside edges add to 36 m, the same as the full 10 m by 8 m rectangle, because the new edges 4 m and 3 m exactly replace the removed 4 m and 3 m. The area is 68 m².
- Press Notch in the middle of an edge. The area is still 68 m², but the perimeter is now 42 m: the two sides of the notch, 3 m each, are extra.
- Make the notch 6 m wide and 5 m deep, in the corner. The perimeter is still 36 m, and the area is 80 − 30 = 50 m².
- Put that 6 m by 5 m notch in the middle of the edge. The perimeter is 36 + 2 × 5 = 46 m.
A rectangular notch cut from a corner does not change the perimeter: the two new edges are as long as the two edges that were removed. A notch cut into the middle of an edge adds two sides, each as long as the notch is deep.
A common mistake: “I split the shape into two rectangles with perimeters 18 m and 14 m, so the whole perimeter is 18 + 14 = 32 m.” Where the two rectangles touch, that edge was counted once in each rectangle but it is inside the shape and not part of the outside. Here the rectangles are 6 m by 3 m and 4 m by 3 m, joined along a 3 m edge: the shape is 10 m by 3 m, and its perimeter is 26 m. Trace the outside of the whole shape with a finger, and add only the edges you pass.
Worked examples
A house end is a rectangle 12 m wide and 5 m high with a triangular roof, 8 m high at the peak, and sloping sides of 10 m. Find its area and its perimeter.
- Split it into the rectangle and the triangle.
- Rectangle: 12 × 5 = 60 m². Triangle: ½ × 12 × 8 = 48 m². Area: 60 + 48 = 108 m².
- Perimeter, outside edges only: the base 12, the two walls 5 and 5, and the two sloping sides 10 and 10. The dashed line is inside and is not counted. 12 + 5 + 5 + 10 + 10 = 42 m.
A flower bed is a trapezium. Its parallel sides are 10 m and 6 m, and they are 4 m apart. One side is upright. Find its area.
- Split it with a dashed line: a rectangle 6 m by 4 m, and a triangle with base 10 − 6 = 4 m and height 4 m.
- Rectangle: 6 × 4 = 24 m². Triangle: ½ × 4 × 4 = 8 m².
- Area: 24 + 8 = 32 m². Check with the formula: ½ × (6 + 10) × 4 = 8 × 4 = 32 m².
Your go. The first two steps are given. The L-shaped patio below is 9 m long and 7 m high overall, with a 4 m by 3 m piece missing from the corner. Find its area and its perimeter.
- The whole rectangle is 9 m by 7 m: 9 × 7 = 63 m².
- The missing piece is 4 m by 3 m: 4 × 3 = 12 m².
- Now take the piece away to find the area, then find the perimeter from the outside edges, then check.
Check the full answer
Area 63 − 12 = 51 m². The notch is in a corner, so the perimeter is the same as the full rectangle's: 9 + 7 + 9 + 7 = 32 m. (Counting the six edges one by one gives 9 + 4 + 4 + 3 + 5 + 7 = 32, too.)
Check before you go on
1. Two squares, each with sides of 4 cm, are joined along a whole side to make a rectangle 8 cm long and 4 cm wide. Each square has a perimeter of 16 cm. What is the perimeter of the rectangle?
Your turn
Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. The L-shaped patio is 12 m long and 7 m high overall, with a 5 m by 3 m piece missing from a corner. What is its area in square metres?
3. The same patio, 12 m by 7 m overall with a 5 m by 3 m piece missing from a corner, needs edging all the way round its outside. How many metres of edging does it need?
Level 2 · Building
4. A house end is a rectangle 8 m wide and 5 m high, with a triangular roof 3 m high at the peak. What is its area in square metres?
5. The same house end has a wall height of 5 m, a width of 8 m, a roof height of 3 m and two sloping roof sides of 5 m each. How far is it round its outside edge, in metres?
6. A paving slab is a trapezium. Its parallel sides are 14 m and 8 m, and they are 4 m apart. The two sloping sides are 5 m each. What is its area in square metres?
Level 3 · Stretch
7. An L-shaped room is made from a rectangle 9 m by 4 m and a second rectangle 3 m wide that stands on one end of it, as shown. The whole room has an area of 57 m². How long is the upright side marked ?, in metres?
8. A square tile has sides of 6 cm. A 2 cm by 2 cm square is cut from one corner, and another 2 cm by 2 cm square is cut from a different corner. What is the perimeter of the tile that is left, in centimetres?
Something to think about
A 10 m by 8 m rectangle has a 4 m by 3 m notch cut from a corner, and its perimeter is still 36 m. Now cut a second notch, 2 m by 2 m, from another corner. What happens to the area, and to the perimeter? Then cut a 2 m by 2 m square out of the middle of the whole shape, not touching any edge. What happens to the perimeter this time, if you count the new inside edge?
One way to think about it
The first notch leaves 68 m², and a second corner notch of 2 m by 2 m takes 4 more away, leaving 64 m². The perimeter is still 36 m, because every corner notch swaps two edges for two equal ones. A hole in the middle adds a new inside edge of 4 × 2 = 8 m, so the edge of the shape, counting the hole, is 44 m. Whether the hole's edge counts depends on the job: fence round the outside is 36 m, but a gardener edging both the bed and the hole needs 44 m.
Exit ticket
9. Rosa finds the area of an L-shaped patio by multiplying its longest length, 9 m, by its longest width, 6 m, and gets 54 m². The missing corner is 4 m by 2 m. What has she done wrong?