GCSE Maths: Compound shapes (Higher)
Part of the GCSE Maths perimeter, area and circles chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Compound shapes
A compound shape (also called a composite shape) is made by joining simple shapes such as rectangles, triangles and trapezia. You already know how to find the area of each simple shape, so the trick is to break the big shape up. There are two ways.
Method 1: split. Draw a line (or lines) to cut the shape into simple pieces, work out the area of each piece, then add.
Method 2: subtract. Imagine the shape is a bigger simple shape with a piece missing. Work out the area of the big shape, work out the area of the missing piece, then subtract.
Finding missing lengths
Diagrams often leave some sides unlabelled. In a shape with only right angles, opposite directions must balance: the horizontal sides along the top add up to the same as the horizontal sides along the bottom, and the same for the vertical sides. So a missing side is a bigger length minus the lengths you already know.
Perimeter
For the perimeter, work out every missing side, then add up all the sides of the outline. Do not include any dotted lines you drew inside the shape. For slanting sides you must be told the length (or find it with Pythagoras, Chapter 20).
Triangles on top
A “house” shape is a rectangle with a triangle on top. The triangle needs the perpendicular height, which is shown by a dotted line up the middle, not the slanting roof sides.
Key ideas
- Split into simple shapes and add the areas, or complete a bigger shape and subtract the missing part.
- Work out any missing lengths first (big total minus the known parts).
- Perimeter = add every side of the outline. Ignore dotted lines inside the shape.
- For a shape with a border, area of border = outer area − inner area. The border is on both sides, so the inner lengths are 2 × (border width) shorter.
- Always write the units: cm² for area, cm for perimeter.
Watch out
A common error is to use the wrong lengths for the small rectangles. If an L-shape is 10 cm wide and its notch is 4 cm wide, the part left on that edge is 10 − 4 = 6 cm, not 10 cm. Write the missing lengths on the diagram before you multiply.Method. Split the shape into two rectangles, find each area, then add.
- Draw a line to split the shape. A vertical cut makes a tall rectangle on the left and a shorter rectangle on the right. Mark the sides of each.Left rectangle: width 10 − 4 = 6 cm, height 8 cm
Right rectangle: width 4 cm, height 8 − 3 = 5 cm - Area of the left rectangle. Area = length × width.6 × 8 = 48 cm²
- Area of the right rectangle. Do the same for the second piece.4 × 5 = 20 cm²
- Add the two areas. The whole shape is made of the two pieces.48 + 20 = 68 cm²
Check. Subtract method: whole rectangle 10 × 8 = 80, take away the notch 4 × 3 = 12: 80 − 12 = 68. ✓
Watch out. Work out the length of each missing side before multiplying. Do not use the labelled lengths of the whole shape for the small pieces.
Method. Imagine the shape is a full rectangle with a corner missing. Find the area of the big rectangle, then subtract the area of the missing piece.
- Area of the big rectangle. Pretend the notch is filled in. The rectangle is W by H.10 × 8 = 80 cm²
- Area of the missing piece. The notch is a small rectangle.4 × 3 = 12 cm²
- Subtract. The shape is the big rectangle without the missing piece.80 − 12 = 68 cm²
Check. Split method: (6 × 8) + (4 × 5) = 48 + 20 = 68. ✓
Watch out. Subtract the missing piece. Adding it would count empty space.
Method. To find the perimeter, first work out every missing side, then add all six sides.
- Find the missing horizontal side. The bottom side W is as long as the top side plus the horizontal part of the notch. So the notch part is the bottom minus the top.10 − 6 = 4 cm
- Find the missing vertical side. The left side H is as long as the lower right side plus the vertical part of the notch. So the notch part is the left minus the lower right.8 − 5 = 3 cm
- Add all six sides. Go round the shape once, ticking each side.10 + 5 + 4 + 3 + 6 + 8 = 36 cm
Check. The steps of the notch add up to the same as the sides they replace, so the perimeter equals the perimeter of the big rectangle: 2 × (10 + 8) = 36. ✓
Watch out. Do not forget the missing sides: an L-shape has six sides, not four.
Method. Split the shape into a rectangle and a triangle. Find the area of each, then add.
- Split the shape. The dotted line splits it into a rectangle at the bottom and a triangle on top.Rectangle: 12 cm by 5 cm
Triangle: base 12 cm, perpendicular height 8 cm - Area of the rectangle. length × width.12 × 5 = 60 cm²
- Area of the triangle. half × base × perpendicular height.½ × 12 × 8 = 96 ÷ 2 = 48 cm²
- Add. The whole shape is the rectangle plus the triangle.60 + 48 = 108 cm²
Check. Check the triangle by using half the base: 6 × 8 = 48. ✓
Watch out. The triangle needs the perpendicular height (the dotted line up the middle), not the slanting side.
Method. The border is the big outer rectangle with the inner rectangle taken away. Find both areas and subtract.
- Area of the outer rectangle. length × width.30 × 22 = 660 cm²
- Find the sides of the inner rectangle. The border is 4 cm wide on both sides, so each inside length is 2 × 4 = 8 cm shorter.30 − 8 = 22 cm
22 − 8 = 14 cm - Area of the inner rectangle. length × width.22 × 14 = 308 cm²
- Subtract. The border is what is left.660 − 308 = 352 cm²
Check. Four strips: two of 30 × 4 = 120 and two of 14 × 4 = 56. 240 + 112 = 352. ✓
Watch out. Take 2 × 4 off each length, not just 4: the border is on both sides.
Method. Write the area of each rectangle using x, add them, and solve the equation.
- Area of A. length × width.x × 6 = 6x
- Area of B. length × width.(x + 3) × 4 = 4x + 12
- Add and set equal to 72. The total area is 72 cm².6x + 4x + 12 = 72
10x + 12 = 72 - Solve. Subtract 12, then divide by 10.10x = 60
x = 6
Check. A = 6 × 6 = 36, B = 9 × 4 = 36, total 72. ✓
Watch out. Expand the bracket (x + 3) × 4 properly: 4x + 12, not 4x + 3.
Exercise 18D Compound shapes
Non-calculatorMethod. Split the shape into two rectangles, find each area, then add.
- Draw a line to split the shape. A vertical cut makes a tall rectangle on the left and a shorter rectangle on the right. Mark the sides of each.Left rectangle: width 9 − 4 = 5 cm, height 6 cm
Right rectangle: width 4 cm, height 6 − 2 = 4 cm - Area of the left rectangle. Area = length × width.5 × 6 = 30 cm²
- Area of the right rectangle. Do the same for the second piece.4 × 4 = 16 cm²
- Add the two areas. The whole shape is made of the two pieces.30 + 16 = 46 cm²
Check. Subtract method: whole rectangle 9 × 6 = 54, take away the notch 4 × 2 = 8: 54 − 8 = 46. ✓
Watch out. Work out the length of each missing side before multiplying. Do not use the labelled lengths of the whole shape for the small pieces.
Method. To find the perimeter, first work out every missing side, then add all six sides.
- Find the missing horizontal side. The bottom side W is as long as the top side plus the horizontal part of the notch. So the notch part is the bottom minus the top.12 − 7 = 5 cm
- Find the missing vertical side. The left side H is as long as the lower right side plus the vertical part of the notch. So the notch part is the left minus the lower right.7 − 4 = 3 cm
- Add all six sides. Go round the shape once, ticking each side.12 + 4 + 5 + 3 + 7 + 7 = 38 cm
Check. The steps of the notch add up to the same as the sides they replace, so the perimeter equals the perimeter of the big rectangle: 2 × (12 + 7) = 38. ✓
Watch out. Do not forget the missing sides: an L-shape has six sides, not four.
Method. Imagine the shape is a full rectangle with a corner missing. Find the area of the big rectangle, then subtract the area of the missing piece.
- Area of the big rectangle. Pretend the notch is filled in. The rectangle is W by H.10 × 8 = 80 cm²
- Area of the missing piece. The notch is a small rectangle.4 × 3 = 12 cm²
- Subtract. The shape is the big rectangle without the missing piece.80 − 12 = 68 cm²
Check. Split method: (6 × 8) + (4 × 5) = 48 + 20 = 68. ✓
Watch out. Subtract the missing piece. Adding it would count empty space.
Method. To find the perimeter, first work out every missing side, then add all six sides.
- Find the missing horizontal side. The bottom side W is as long as the top side plus the horizontal part of the notch. So the notch part is the bottom minus the top.10 − 6 = 4 cm
- Find the missing vertical side. The left side H is as long as the lower right side plus the vertical part of the notch. So the notch part is the left minus the lower right.8 − 5 = 3 cm
- Add all six sides. Go round the shape once, ticking each side.10 + 5 + 4 + 3 + 6 + 8 = 36 cm
Check. The steps of the notch add up to the same as the sides they replace, so the perimeter equals the perimeter of the big rectangle: 2 × (10 + 8) = 36. ✓
Watch out. Do not forget the missing sides: an L-shape has six sides, not four.
Method. Fill in the gap to make a full rectangle, find its area, then subtract the area of the gap.
- Area of the full rectangle. Imagine the gap is filled in, so the shape is W by H.14 × 8 = 112 cm²
- Area of the gap. The gap is a small rectangle with the labelled width and depth.6 × 5 = 30 cm²
- Subtract. The shape is the full rectangle with the gap taken out.112 − 30 = 82 cm²
Check. Split into three rectangles: left 4 × 8 = 32, bottom 6 × 3 = 18, right 4 × 8 = 32. Total 82. ✓
Watch out. Subtract the gap: do not add it.
Method. Split the shape into a rectangle and a triangle. Find the area of each, then add.
- Split the shape. The dotted line splits it into a rectangle at the bottom and a triangle on top.Rectangle: 12 cm by 5 cm
Triangle: base 12 cm, perpendicular height 8 cm - Area of the rectangle. length × width.12 × 5 = 60 cm²
- Area of the triangle. half × base × perpendicular height.½ × 12 × 8 = 96 ÷ 2 = 48 cm²
- Add. The whole shape is the rectangle plus the triangle.60 + 48 = 108 cm²
Check. Check the triangle by using half the base: 6 × 8 = 48. ✓
Watch out. The triangle needs the perpendicular height (the dotted line up the middle), not the slanting side.
Method. Perimeter is the distance all the way round the outside. Add the outside sides only: the dotted lines are inside the shape.
- List the outside sides. Bottom, two vertical sides and the two sloping sides of the roof. The two sloping sides are equal.12, 5, 5, 10, 10
- Add them. The perimeter is the total.12 + 5 + 5 + 10 + 10 = 42
- Write the units. Perimeter is a length.Perimeter = 42 cm
Check. Group the equal sides: 12 + 2 × 5 + 2 × 10 = 12 + 10 + 20 = 42. ✓
Watch out. Do not include the dotted line across the middle or the dotted height.
Method. The shape is a rectangle with a triangle cut off one corner. Find the area of the whole rectangle (the dotted lines complete it) and subtract the triangle.
- Find the sides of the missing triangle. Its base is the top of the rectangle minus the top of the shape. Its height is the side of the rectangle minus the right-hand side of the shape.base = 12 − 8 = 4 cm
height = 9 − 6 = 3 cm - Area of the whole rectangle. length × width.12 × 9 = 108 cm²
- Area of the missing triangle. half × base × height.½ × 4 × 3 = 6 cm²
- Subtract. The shape is the rectangle without the triangle.108 − 6 = 102 cm²
Check. Split instead: rectangle 8 × 9 = 72 plus a trapezium with parallel sides 9 and 6, width 4: ½ × 15 × 4 = 30. Total 102. ✓
Watch out. The dotted lines are not part of the shape, but they help you see the rectangle.
Method. Add up all five sides of the outline. The cut-off side is given as 5 cm.
- List the five sides. Go round once: bottom, right side, the slanting cut, top, left side.12, 9 − 3 = 6, 5, 12 − 4 = 8, 9
- Add them. The perimeter is the total.12 + 6 + 5 + 8 + 9 = 40 cm
Check. Add in a different order: 9 + 12 + 8 + 5 + 6 = 40. ✓
Watch out. The dotted lines are not part of the shape, so they are not in the perimeter.
Method. The border is the big outer rectangle with the inner rectangle taken away. Find both areas and subtract.
- Area of the outer rectangle. length × width.30 × 22 = 660 cm²
- Find the sides of the inner rectangle. The border is 4 cm wide on both sides, so each inside length is 2 × 4 = 8 cm shorter.30 − 8 = 22 cm
22 − 8 = 14 cm - Area of the inner rectangle. length × width.22 × 14 = 308 cm²
- Subtract. The border is what is left.660 − 308 = 352 cm²
Check. Four strips: two of 30 × 4 = 120 and two of 14 × 4 = 56. 240 + 112 = 352. ✓
Watch out. Take 2 × 4 off each length, not just 4: the border is on both sides.
Method. The path is the big rectangle (lawn + path) with the lawn taken away. The path is on both sides, so the outer lengths are 2 × 1 m longer than the lawn.
- Find the outer rectangle. The path adds 1 m on each side, so 2 m to each length.12 + 2 = 14 m and 8 + 2 = 10 m
- Area of the outer rectangle. length × width.14 × 10 = 140 m²
- Area of the lawn. length × width.12 × 8 = 96 m²
- Subtract. The path is what is left.140 − 96 = 44 m²
Check. Four strips: two of 14 × 1 = 14 and two of 8 × 1 = 8. 14 + 14 + 8 + 8 = 44. ✓
Watch out. Add 2 × 1 m to each length (the path is on both sides), not just 1 m.
Method. Find the area of the floor, then multiply by the cost of each square metre.
- Area of the whole 8 m by 6 m rectangle. Imagine the corner filled in.8 × 6 = 48 m²
- Area of the missing corner. The notch is 3 m by 2 m.3 × 2 = 6 m²
- Area of the floor. Subtract the missing corner.48 − 6 = 42 m²
- Cost. Each square metre costs £22.42 × 22 = 924
Cost = £924
Check. Split method: (8 − 3) × 6 = 30 and 3 × (6 − 2) = 12; 30 + 12 = 42 m². ✓
Watch out. Cost uses area. Do not use the perimeter for flooring.
Method. Find the perimeter of the garden, then multiply by the cost per metre.
- Find the missing sides. Bottom minus top gives the horizontal notch side; left minus right gives the vertical notch side.15 − 9 = 6 m and 10 − 6 = 4 m
- Add all six sides. Go all the way round.15 + 6 + 6 + 4 + 9 + 10 = 50 m
- Multiply by the cost per metre. Each metre costs £12.50 × 12 = £600
Check. The perimeter equals that of the 15 m by 10 m rectangle: 2 × (15 + 10) = 50 m. ✓
Watch out. Fencing goes round the edge: use the perimeter.
Method. Look at what 10 × 8 and 4 × 3 represent.
- Explain the mistake. 10 × 8 is the area of the whole rectangle, including the missing corner. The 4 by 3 corner is not part of the shape, so it must be taken away, not added.Leo should have used 10 × 8 − 4 × 3.
- Correct the working. Subtract the area of the missing corner.80 − 12 = 68 cm²
Check. Split method: 6 × 8 = 48 and 4 × 5 = 20; 48 + 20 = 68 cm². ✓
Watch out. A missing corner is taken away.
Method. Split the shape into a left-hand rectangle and a right-hand rectangle, then add their areas.
- Find the missing sides. The left rectangle is as wide as the top edge. The right rectangle is as wide as the notch and as tall as the right side.Left: width 9 − 3 = 6, height 7
Right: width 3, height 7 − 2 = 5 - Area of the left rectangle. length × width.6 × 7 = 42 cm²
- Area of the right rectangle. length × width.3 × 5 = 15 cm²
- Add. The shape is both rectangles together.42 + 15 = 57 cm²
- Conclude. State what you have shown.The area is 57 cm², as required.
Check. Subtract method: 9 × 7 = 63, then 63 − 3 × 2 = 63 − 6 = 57. ✓
Watch out. End a “show that” question with a sentence.
Method. Write the area of each rectangle with x, add them and solve.
- Area of A. length × width.6x
- Area of B. Expand the bracket.4(x + 3) = 4x + 12
- Make an equation. The total area is 72.10x + 12 = 72
- Solve. Subtract 12 and divide by 10.10x = 60
x = 6
Check. 36 + 36 = 72. ✓
Watch out. Expand the bracket properly: 4(x + 3) = 4x + 12.
Method. Add all six sides of the outline.
- List the sides. Start at the bottom and go anticlockwise.3x, x, x, x, 2x, 2x
- Add the terms. Count the x’s: 3 + 1 + 1 + 1 + 2 + 2 = 10.3x + x + x + x + 2x + 2x = 10x
Check. The perimeter of the full 3x by 2x rectangle is 2(3x + 2x) = 10x, the same. ✓
Method. Solve 10x = 50 first, then work out the area by subtracting.
- Find x. Divide both sides by 10.10x = 50, so x = 5
- Find the big rectangle. Sides 3x and 2x.15 cm by 10 cm = 150 cm²
- Subtract the square. It is x by x.5 × 5 = 25 cm²
150 − 25 = 125 cm²
Check. Algebra: 6x2 − x2 = 5x2 = 5 × 25 = 125. ✓