Surface Area and Volume of 3D Shapes – IGCSE Formula Guide

Why This Topic Carries Heavy Marks

Mensuration questions — surface area and volume of 3D shapes — regularly appear on both Paper 2 and Paper 4. They often carry 5 to 8 marks per question and frequently combine with other topics like Pythagoras’ theorem, trigonometry, or algebra. The formulas for spheres and cones are given on the formula sheet, but you still need to know how to use them correctly.

Students who practise these questions thoroughly often pick up reliable marks because the method, once learned, is very repeatable.

The Formulas You Need

Cuboid (Rectangular Prism)

• Volume = length × width × height = lwh
• Surface area = 2(lw + lh + wh)

Cylinder

• Volume = πr²h
• Curved surface area = 2πrh
• Total surface area = 2πrh + 2πr² (curved surface + two circular ends)

Cone

• Volume = (1/3)πr²h
• Curved surface area = πrl (where l is the slant height)
• Total surface area = πrl + πr² (curved surface + circular base)

The slant height l, the radius r, and the vertical height h are related by Pythagoras: l² = r² + h².

Sphere

• Volume = (4/3)πr³
• Surface area = 4πr²

Hemisphere

• Volume = (2/3)πr³
• Curved surface area = 2πr²
• Total surface area = 2πr² + πr² = 3πr² (curved surface + flat circular face)

Pyramid

• Volume = (1/3) × base area × height

The surface area of a pyramid depends on the shape of the base and the triangular faces. You usually calculate each face separately and add them up.

Worked Examples

Worked Example 1: Cylinder

A cylinder has radius 5 cm and height 12 cm. Find its volume and total surface area.

Volume = πr²h = π × 5² × 12 = π × 25 × 12 = 300π = 942.5 cm³ (to 1 d.p.)

Total surface area = 2πrh + 2πr² = 2π(5)(12) + 2π(5²) = 120π + 50π = 170π = 534.1 cm² (to 1 d.p.)

Worked Example 2: Cone

A cone has base radius 6 cm and vertical height 8 cm. Find the volume and the curved surface area.

Volume = (1/3)πr²h = (1/3) × π × 36 × 8 = 96π = 301.6 cm³ (to 1 d.p.)

First find the slant height: l = √(r² + h²) = √(36 + 64) = √100 = 10 cm.

Curved surface area = πrl = π × 6 × 10 = 60π = 188.5 cm² (to 1 d.p.)

Worked Example 3: Sphere

A sphere has diameter 14 cm. Find its volume and surface area.

Radius = 14 ÷ 2 = 7 cm.

Volume = (4/3)πr³ = (4/3) × π × 343 = (1372/3)π = 1436.8 cm³ (to 1 d.p.)

Surface area = 4πr² = 4 × π × 49 = 196π = 615.8 cm² (to 1 d.p.)

Worked Example 4: Combined Shape

A solid is made of a cylinder with a hemisphere on top. The cylinder has radius 4 cm and height 10 cm. The hemisphere has the same radius. Find the total volume and the total external surface area.

Volume of cylinder = π × 16 × 10 = 160π

Volume of hemisphere = (2/3) × π × 64 = (128/3)π

Total volume = 160π + (128/3)π = (480/3 + 128/3)π = (608/3)π = 636.7 cm³ (to 1 d.p.)

Surface area:

• Curved surface of cylinder = 2π(4)(10) = 80π
• Base of cylinder (bottom circle) = π(4²) = 16π
• Curved surface of hemisphere = 2π(4²) = 32π
• Note: the top circle of the cylinder is not exposed (the hemisphere sits on it)

Total surface area = 80π + 16π + 32π = 128π = 402.1 cm² (to 1 d.p.)

Worked Example 5: Working Backwards

A sphere has a volume of 500 cm³. Find its radius.

(4/3)πr³ = 500

r³ = 500 × 3/(4π) = 1500/(4π) = 375/π

r³ = 119.366…

r = ³√119.366… = 4.92 cm (to 3 s.f.)

Exam Tips for Mensuration Questions

1. Read carefully for diameter vs. radius. A very common error is using the diameter where the formula requires the radius. If the question says “diameter 10 cm,” your radius is 5 cm.

2. Check the formula sheet. The formulas for sphere volume, sphere surface area, cone volume, and cone curved surface area are printed on the formula sheet at the front of your exam paper. Use them — do not try to memorise them under pressure.

3. Show the substitution step. Write the formula, then show the numbers substituted in. This earns method marks even if you make an arithmetic error.

4. Watch your units. Volume is in cubic units (cm³, m³). Surface area is in square units (cm², m²). If the question asks for litres, remember 1 litre = 1000 cm³.

5. Combined shapes require you to identify which surfaces are exposed. When a hemisphere sits on top of a cylinder, the flat face of the hemisphere and the top circle of the cylinder are both hidden.

6. Leave answers in terms of π if the question says “give your answer in terms of π.” Otherwise, use your calculator and round as instructed.

Common Mistakes

• Forgetting to use (1/3) for cones and pyramids — their volume is one third of the corresponding prism.
• Using height instead of slant height in the cone curved surface area formula.
• Not squaring or cubing the radius correctly, especially in sphere questions.
• Forgetting the base when calculating total surface area of a cone or cylinder.

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