Before you begin
What you need: Plane areas, squares, cubes and unit conversion.
- Calculate volumes of common solids
- Distinguish curved and total surface area
- Convert between cubic units and capacity
Understand the idea
Volume measures occupied space
A prism or cylinder has the same cross-sectional area throughout its perpendicular height. Its volume equals base area × height: stacking equal layers builds the solid.
Surface area measures the outside
A cuboid has three pairs of equal rectangular faces. A closed cylinder has a curved surface and two circular ends. An open container may have fewer surfaces to count.
Similar shapes scale in three dimensions
Scaling all lengths by k multiplies surface area by k² and volume by k³. Capacity units connect to volume: 1 L = 1000 cm³ and 1 m³ = 1000 L.
Stack the circular base
A method you can reuse
Cuboid V = lbh; cylinder V = πr²h; cone V = ⅓πr²h
Cylinder curved area = 2πrh; closed total area = 2πrh + 2πr². Sphere V = 4πr³/3.- Decide whether the question asks for capacity, volume or covering area.
- Identify the solid, label radius and perpendicular height, and choose the required faces.
- Calculate in consistent units, then convert capacity if needed.
A cone and cylinder with the same base radius and perpendicular height have volumes in the ratio 1:3.
Worked examples
Read the question first. Try a step yourself, then compare your reasoning.
A box's capacity
Find the volume of a 5 cm by 4 cm by 3 cm cuboid.
- Base area = 5 × 4 = 20 cm².
- Multiply by height: 20 × 3.
- Volume uses cubic units.
Cylinder volume
A cylinder has radius 7 cm and height 10 cm. Use π = 22/7.
- Base area = (22/7) × 7² = 154 cm².
- Volume = 154 × 10 = 1540 cm³.
- For capacity in litres, divide by 1000.
Cover an open container
Find the surface area of a cylinder open at the top, radius 3 cm, height 5 cm.
- Curved area = 2π × 3 × 5 = 30π cm².
- There is one base: π × 3² = 9π cm².
- Add only the surfaces present.
Check your understanding
Quick questions, with explanations. These are for self-study and do not affect your account score.
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