Area, volume and surface area

Maths · Geometry and measures. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Area measures the flat space inside a 2D shape, in square units (cm²). Volume measures the space inside a 3D solid, in cubic units (cm³). Surface area is the total area of all the faces of a solid.

A prism has the same cross-section all the way along, so its volume is the area of that cross-section times its length. A cylinder is a prism with a circular cross-section. Pyramids and cones come to a point, so their volume is one third of the matching prism.

A sector is a fraction of a circle. The fraction is the angle divided by 360, and it works for both the arc length and the area. Leave π in the answer when an exact value is asked for.

Rules to remember

Area of a triangle = ½ × base × perpendicular height; area of a parallelogram = base × perpendicular height.
Area of a trapezium = ½ × (a + b) × h, where a and b are the parallel sides and h is the distance between them.
Circle: circumference = π × diameter (C = πd = 2πr); area = π × radius² (A = πr²).
Volume of a prism = area of cross-section × length; volume of a cylinder = πr²h.
Volume of a pyramid or cone = (1/3) × base area × vertical height; cone: V = (1/3)πr²h, curved surface area = πrl (l = slant height).
Sphere: volume = (4/3)πr³; surface area = 4πr².
Arc length = (θ/360) × 2πr; sector area = (θ/360) × πr², with θ in degrees.

Common mistake

Using the diameter where the formula needs the radius, or using a slant length where the formula needs the perpendicular (vertical) height. Halve the diameter first and check which height you have been given.

Worked example

A solid cone has base radius 6 cm and vertical height 8 cm. Find its exact total surface area.

Slant height: l = √(6² + 8²) = √(36 + 64) = √100 = 10 cm.
Curved surface area = πrl = π × 6 × 10 = 60π cm².
Base area = πr² = π × 6² = 36π cm².
Total = 60π + 36π = 96π cm².

Answer: 96π cm²

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