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This lesson covers DfE content statements L2.17, L2.20 and L2.21 — calculating the volume and surface area of cuboids, prisms and cylinders, and understanding 2-D representations of 3-D objects (including plans and elevations).
Volume is the amount of 3-D space a shape occupies. It is measured in cubic units (cm³, m³, etc.).
A cuboid is a 3-D shape where every face is a rectangle (a box shape).
Volume = length × width × height
Scenario: A garden raised bed is 2.4 m long, 1.2 m wide, and 0.3 m deep. How much soil is needed to fill it?
Volume = 2.4 × 1.2 × 0.3 = 0.864 m³
If soil is sold in 50-litre bags, how many bags are needed?
1 m³ = 1,000 litres → 0.864 m³ = 864 litres 864 ÷ 50 = 17.28 → 18 bags (round up)
Scenario: A shipping container measures 12 m × 2.4 m × 2.6 m. What is its volume?
Volume = 12 × 2.4 × 2.6 = 74.88 m³
A prism is any 3-D shape with a constant cross-section along its length. A cuboid is a special case of a prism.
Volume of a prism = Area of cross-section × length
graph LR
A["Find the area of<br/>the cross-section<br/>(the end face)"] --> B["Multiply by the<br/>length (depth)<br/>of the prism"]
B --> C["Volume =<br/>Cross-section area × Length"]
Scenario: A loft has a triangular cross-section with a base of 8 m and a height of 3 m. The loft is 12 m long. What is the volume of the loft space?
Cross-section area = 1/2 × 8 × 3 = 12 m² Volume = 12 × 12 = 144 m³
Scenario: A drainage channel has a trapezoidal cross-section with parallel sides 1.5 m and 0.8 m, height 0.6 m, and length 25 m. What is its capacity?
Cross-section area = 1/2 × (1.5 + 0.8) × 0.6 = 1/2 × 2.3 × 0.6 = 0.69 m² Volume = 0.69 × 25 = 17.25 m³
Convert to litres: 17.25 × 1,000 = 17,250 litres
A cylinder is a prism with a circular cross-section.
Volume = π × r² × h
Scenario: A cylindrical water tank has a radius of 0.5 m and a height of 1.2 m. How many litres of water can it hold?
Volume = π × 0.5² × 1.2 = π × 0.25 × 1.2 = π × 0.3 = 0.9425 m³
Convert to litres: 0.9425 × 1,000 = 942.5 litres
Scenario: A cylindrical concrete pillar has a diameter of 40 cm and a height of 3 m. What is its volume in cm³?
Radius = 40 ÷ 2 = 20 cm; Height = 300 cm Volume = π × 20² × 300 = π × 400 × 300 = 376,991 cm³ (to the nearest cm³)
Or in m³: π × 0.2² × 3 = 0.377 m³
Exam Tip: Always check the units in the question. If the radius is given in centimetres and the height in metres, you MUST convert to the same unit before calculating. Mixing units is a very common error.
Surface area is the total area of all the faces of a 3-D shape. It is measured in square units (cm², m²).
A cuboid has 6 rectangular faces (3 pairs of identical faces).
Surface area = 2(lw + lh + wh)
Scenario: A gift box measures 30 cm × 20 cm × 15 cm. What is the minimum amount of wrapping paper needed?
SA = 2(30×20 + 30×15 + 20×15) = 2(600 + 450 + 300) = 2 × 1,350 = 2,700 cm²
A cylinder has two circular ends and a curved surface that, when "unrolled," forms a rectangle.
Surface area = 2πr² + 2πrh
(Two circles + one rectangle of width 2πr and height h)
Scenario: A cylindrical oil tank has a radius of 1.5 m and a height of 3 m. What area needs to be painted (the curved surface and the top — the bottom sits on the ground)?
Curved surface = 2π × 1.5 × 3 = 28.27 m² Top circle = π × 1.5² = 7.07 m² Total = 28.27 + 7.07 = 35.34 m²
| Volume unit | Capacity equivalent |
|---|---|
| 1 cm³ | 1 ml |
| 1,000 cm³ | 1 litre |
| 1 m³ | 1,000 litres |
Scenario: A rectangular swimming pool is 25 m long, 10 m wide, and has an average depth of 1.5 m. How many litres of water does it hold?
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