Area, Perimeter & Volume

Area of triangles and composite figures. Volume of cubes and cuboids — including working backwards to find an unknown dimension given volume and other sides.

area of trianglecomposite areafind unknown dimension P2 · calc

Area of triangle = ½ × base × height Volume of cuboid = l × w × h Volume of cube = side³ 1 litre = 1000 cm³

Part A — Area of triangles

Question 1 — Short answer (2 marks) Paper 1 · no calculator Area of triangle

Find the area of the triangle below.

FormulaArea = ½ × base × height

Base= 14 cm, Height = 10 cm

Area= ½ × 14 × 10 = 70 cm²

Area = 70 cm²

The height must be perpendicular (90°) to the base — shown by the dashed line and the right-angle box. Never use a slanted side as the height.

Question 2 — Short answer (2 marks) Paper 1 · no calculator Composite area · triangle + rectangle

The figure below is made up of a rectangle and a triangle. Find the total area of the figure.

Rectangle= 16 × 8 = 128 cm²

Triangle= ½ × 16 × 5 = 40 cm²

Total= 128 + 40 = 168 cm²

Total area = 168 cm²

The triangle sits on top of the rectangle, so its base = the rectangle's width (16 cm). The height of the triangle (5 cm) is only the triangular portion — do not add 8 cm to it.

Part B — Volume of cuboids and cubes

Question 3 — Short answer (2 marks) Paper 2 · calculator Volume · liquid in tank

A rectangular tank is 30 cm long, 20 cm wide and 25 cm tall. It is filled with water to a height of 18 cm. How many litres of water are in the tank?

Volume= l × w × h of water = 30 × 20 × 18

= 600 × 18 = 10 800 cm³

Convert= 10 800 ÷ 1000 = 10.8 litres

10.8 litres of water

Common trap: Using 25 cm (the full tank height) instead of 18 cm (the water height). Always use the height of the water, not the tank, when finding volume of liquid.

Part C — Finding unknown dimensions (reverse problems)

Question 4 — Structured (3 marks) Paper 2 · calculator Find unknown height

A cuboid has a base area of 48 cm² and a volume of 336 cm³. What is the height of the cuboid?

FormulaVolume = base area × height

RearrangeHeight = Volume ÷ base area

Height= 336 ÷ 48 = 7 cm

Height = 7 cm

Volume = l × w × h = (l × w) × h = base area × height. When base area is given directly, simply divide volume by base area to find height.

Question 5 — Long answer (4–5 marks) Paper 2 · calculator Water transfer · find new height Distinction level

Tank A measures 20 cm × 15 cm × 30 cm and is completely full of water. All the water from Tank A is poured into Tank B, which measures 25 cm × 12 cm × 40 cm and is empty.

(a) What is the volume of water?
(b) What is the height of the water in Tank B?
(c) How many more cm must Tank B be filled to reach the top?

(a) Volume= 20 × 15 × 30 = 9000 cm³

(b) Base B= 25 × 12 = 300 cm²

Height= 9000 ÷ 300 = 30 cm

(a) 9000 cm³ | (b) 30 cm | (c) 10 cm more

Check before answering (b): Does 30 cm exceed Tank B's height of 40 cm? No — so the water fits. Always verify the water height is less than the tank height, or it means the tank overflows and the question has an error.