Area of Compound Shapes

Find each rectangle’s area before adding.

Find the complete rectangle and subtract the missing corner.

The hole’s area must be removed from the sheet’s area.

Use different area formulae for the rectangle and triangle.

The missing height and the lower strip together make the whole height.

You can subtract the missing corner or split the L-shape into two rectangles.

1. Outside rectangle: 10\times7=70\text{ m}^2.
2. Missing corner: 6\times4=24\text{ m}^2.
3. L-shape: 70-24=46\text{ m}^2.

Subtract the opening from the outside rectangle.

1. Outside area: 12\times10=120\text{ cm}^2.
2. Opening area: 8\times6=48\text{ cm}^2.
3. Frame area: 120-48=72\text{ cm}^2.

There are two triangular areas as well as the rectangle.

1. Rectangle: 5\times8=40\text{ cm}^2.
2. One triangle: \frac12\times5\times4=10\text{ cm}^2.
3. Total: 40+2\times10=60\text{ cm}^2.

The missing hole is part of the original rectangle.

1. The hole must be subtracted once from the whole rectangle.
2. 50-9=41\text{ cm}^2.

Calculate both rectangular areas using the decimal height.

1. Outside: 5\times4=20\text{ m}^2.
2. Corner: 2\times1.5=3\text{ m}^2.
3. Remaining: 20-3=17\text{ m}^2.

Find the triangle’s height by subtracting the rectangular height from the total height.

1. Triangle height: 5-3=2 m.
2. Rectangle: 7\times3=21\text{ m}^2. Triangle: \frac12\times7\times2=7\text{ m}^2.
3. Total: 21+7=28\text{ m}^2.

Subtract the window area from the complete wall area.

1. Wall: 5\times2.4=12\text{ m}^2.
2. Window: 1.5\times1.2=1.8\text{ m}^2.
3. Area to paint: 12-1.8=10.2\text{ m}^2.

Find the L-shaped area before applying the price per square metre.

1. Complete the room to a 8\times6 rectangle. The hatched upper-right corner is missing: it is 3\times2.

2. Whole rectangle: 8\times6=48\text{ m}^2. Missing corner: 3\times2=6\text{ m}^2.
3. Room area: 48-6=42\text{ m}^2.
4. Cost: 42\times14=588, so the carpet costs £588.

Halve the diameter to get the radius, then subtract the circular area.

1. Garden area: 10\times6=60\text{ m}^2.
2. Radius: 2\div2=1 m, so flowerbed area is \pi\times1^2=\pi\text{ m}^2.
3. Lawn area: 60-\pi=56.8584\ldots\text{ m}^2, or 56.9\text{ m}^2 to 1 decimal place.

Convert the cut-out dimensions to centimetres first.

1. 0.3\text{ m}=30 cm and 0.2\text{ m}=20 cm.
2. Whole card: 120\times80=9600\text{ cm}^2. Cut-out: 30\times20=600\text{ cm}^2.
3. Remaining area: 9600-600=9000\text{ cm}^2.

Find the terrace area, divide by the coverage and check whether rounding up is needed.

1. Complete the terrace to a 11\times8 rectangle. The hatched upper-right corner is missing: it is 4\times3.

2. Whole rectangle: 11\times8=88\text{ m}^2. Missing corner: 4\times3=12\text{ m}^2.
3. Terrace area: 88-12=76\text{ m}^2.
4. 76\div6=12\frac23.
5. Twelve bags cover only 72\text{ m}^2, so 13 whole bags are needed.

Find the missing area first, then undo multiplying by the known height.

1. Full area: 12\times7=84\text{ cm}^2.
2. Missing area: 84-64=20\text{ cm}^2.
3. Missing width: 20\div4=5 cm.

The path adds its width on both ends of each dimension.

1. Outside length: 6+1+1=8 m. Outside width: 4+1+1=6 m.
2. Outside area: 8\times6=48\text{ m}^2. Pond area: 6\times4=24\text{ m}^2.
3. Path area: 48-24=24\text{ m}^2.

Find the area of the surrounding rectangle and subtract the four corner triangles.

1. The rectangle is (5+9)=14 cm wide and (6+4)=10 cm high, so its area is 14\times10=140\text{ cm}^2.
2. The top-left and bottom-right triangles each have area \frac12\times5\times6=15\text{ cm}^2.
3. The other two triangles each have area \frac12\times9\times4=18\text{ cm}^2.
4. The shaded area is 140-2\times15-2\times18=74\text{ cm}^2.

Use the total height to find the triangle’s height. Find the front area, remove the opening and allow for two coats.

1. The triangular part has height 5-3=2 m.
2. Full front: 8\times3+\frac12\times8\times2=24+8=32\text{ m}^2.
3. Painted surface: 32-2\times1=30\text{ m}^2.
4. Two coats require coverage of 2\times30=60\text{ m}^2.
5. 60\div12=5, so exactly 5 tins are needed.