IGCSE Maths area, volume and surface area: sectors, cones, spheres and frustums

Which formulas are given, and which you must learn

From the Cambridge 0580 List of formulas (Core and Extended) and Edexcel 4MA1 Appendices 4 and 5.

The key ideas

An arc is a fraction of the circumference and a sector is the same fraction of the area. With angle θ, arc length = (θ/360) × 2πr and sector area = (θ/360) × πr^2. The perimeter of a sector is the arc plus two radii, which students often forget.

A prism has the same cross-section all the way through, so its volume is cross-sectional area × length. Cylinders are circular prisms. Pyramids and cones hold one third of the matching prism: V = 1/3 × base area × height. A sphere has V = 4/3 πr^3 and surface area 4πr^2; a hemisphere is half the volume, and its total surface area is half the sphere plus the flat circle.

For a cone, the curved surface uses the slant height l, not the vertical height: curved surface area = πrl, where l = sqrt(r^2 + h^2) by Pythagoras. A frustum (Cambridge Extended) is a cone with its top cut off: subtract the small cone from the large one.

Units scale with the dimension: 1 m = 100 cm, so 1 m^2 = 100^2 = 10 000 cm^2 and 1 m^3 = 1 000 000 cm^3. One litre is 1000 cm^3, so 1 m^3 = 1000 litres. Answers may be asked for 'in terms of π', which means leave π in the answer, for example 24π.

Worked example 1: a sector

1. A sector has radius 8 cm and angle 135°. Find its arc length, perimeter and area.
2. Fraction of the circle: 135/360 = 0.375.
3. Arc length = 0.375 × 2π × 8 = 6π = 18.8 cm (3 s.f.).
4. Perimeter = arc + two radii = 6π + 16 = 34.8 cm.
5. Area = 0.375 × π × 8^2 = 24π = 75.4 cm^2. Check: the whole circle is 64π, and 24π is three eighths of it, matching 135/360.

Worked example 2: a cone

1. A cone has base radius 6 cm and vertical height 8 cm. Find its volume and total surface area, in terms of π.
2. Volume = 1/3 × π × 6^2 × 8 = 1/3 × 288π = 96π cm^3 (about 302 cm^3).
3. Slant height l = sqrt(6^2 + 8^2) = sqrt(100) = 10 cm.
4. Curved surface = πrl = π × 6 × 10 = 60π. Base = π × 6^2 = 36π.
5. Total surface area = 60π + 36π = 96π cm^2 (about 302 cm^2). The numbers happen to match the volume here, but the units are different.

Worked example 3 (Cambridge Extended): a frustum and a unit conversion

1. A cone of radius 6 cm and height 12 cm has a cone of radius 2 cm and height 4 cm cut from its top. Find the volume of the frustum.
2. Large cone: 1/3 × π × 36 × 12 = 144π. Small cone: 1/3 × π × 4 × 4 = 16π/3.
3. Frustum = 144π - 16π/3 = 432π/3 - 16π/3 = 416π/3 = 435.6..., so 436 cm^3 (3 s.f.).
4. Unit conversion: a tank measures 1.2 m by 0.8 m by 0.5 m. Its volume is 0.48 m^3, which is 0.48 × 1000 = 480 litres.

Common mistakes that cost marks

• Using the vertical height instead of the slant height in πrl.
• Forgetting the two radii in the perimeter of a sector, or the flat circle on a hemisphere.
• Converting area units with a factor of 100 instead of 10 000 (cm^2 to m^2).
• Using diameter where the formula needs radius.
• Rounding π or intermediate values early; use the calculator's π or 3.142 on Cambridge papers.
• Missing faces when finding the surface area of a prism; list them before adding.

Exam technique and how a tutor helps

Read the first page of the paper: the formula list is there, and knowing what is given saves memorising what you do not need. Cambridge does not give the trapezium formula; Edexcel does. Neither gives arc or sector formulas. When the question says 'in terms of π', keep π as a symbol throughout. On non-calculator papers, expect numbers like radius 6 or angle 120° that simplify cleanly.

For composite shapes, sketch the pieces separately and label each one with its own dimensions before you calculate. Many multi-step questions mix units, such as a length in cm and a capacity in litres, so convert before you calculate.

In one-to-one lessons a tutor can sketch nets and cross-sections on the shared whiteboard, so the student sees which faces or pieces make up a surface area. Lessons then move to the multi-step problems that appear late on papers, such as filling a cylinder from a cone, where the skill is planning the steps rather than recalling a formula.

Self-check: can you do these?

• Find the area of a trapezium with parallel sides 5 cm and 9 cm and height 4 cm. (Answer: 28 cm^2)
• Find the volume of a sphere of radius 3 cm. (Answer: 36π = 113 cm^3)
• Find the circumference of a circle of diameter 10 cm. (Answer: 31.4 cm)
• Find the volume of a cylinder with radius 2 cm and height 5 cm. (Answer: 20π = 62.8 cm^3)
• Convert 3.5 m^2 to cm^2. (Answer: 35 000 cm^2)