Geometry and Measurement
The Geometry and Measurement domain tests spatial reasoning, angle and circle facts, similarity and congruence, Pythagoras’ theorem, and the measurement of length, area, surface area and volume. Questions often combine a diagram-free description with a unit conversion or a multi-step calculation.
Core rules and concepts
- The interior angle sum of an n-sided polygon is (n − 2) × 180°. In a regular polygon each exterior angle is 360°/n.
- Vertically opposite angles are equal; angles on a straight line sum to 180°; angles at a point sum to 360°.
- For similar figures, corresponding angles are equal and corresponding lengths are in a constant ratio. Areas scale by the square of that ratio; volumes scale by its cube.
- Pythagoras’ theorem applies only to right-angled triangles: a² + b² = c², where c is the hypotenuse.
- Circle: circumference = 2πr, area = πr². A sector with angle θ has arc length (θ/360) × 2πr and area (θ/360) × πr².
- Volume of a prism or cylinder = base area × height. Convert units before calculating: 1 m = 100 cm, so 1 m² = 10,000 cm² and 1 m³ = 1,000,000 cm³.
Worked example
A sector has radius 10 cm and angle 72°. Its arc length is (72/360) × 2π × 10 = 0.2 × 20π = 4π cm ≈ 12.57 cm. Its area is (72/360) × π × 10² = 0.2 × 100π = 20π cm² ≈ 62.83 cm². The fraction 72/360 = 1/5 is the key: the sector is one-fifth of the whole circle.
Traps that cost marks
- Using the diameter in place of the radius in circle formulas.
- Applying a length scale factor directly to area or volume.
- Mixing units, such as multiplying centimetres by metres without converting.
- Assuming a triangle is right-angled because it looks so in a description.
- Adding surface areas of faces that are not exposed.