Geometry and Mensuration
Geometry and mensuration combines angle reasoning with measurement of length, area and volume. Expect questions on parallel-line angles, polygons, circle theorems, similarity and congruence, Pythagoras, and the surface area and volume of prisms, cylinders, cones, spheres and pyramids.
Core rules
- Angles on a straight line sum to 180°; angles at a point sum to 360°; alternate and corresponding angles between parallel lines are equal, and co-interior angles sum to 180°.
- Interior angle sum of an n-sided polygon = (n − 2) × 180°; exterior angles sum to 360°.
- Circle theorems: the angle at the centre is twice the angle at the circumference; the angle in a semicircle is 90°; opposite angles of a cyclic quadrilateral are supplementary; a tangent is perpendicular to the radius; alternate segment angles are equal.
- Similar figures: lengths scale by k, areas by k², volumes by k³.
- Sector: arc = (θ/360) × 2πr, area = (θ/360) × πr².
- Cone: l = √(r² + h²), curved area = πrl, volume = (1/3)πr²h.
Worked example
A cone has radius 7 cm and height 24 cm. Using π = 22/7, find its slant height, volume and curved surface area. Reasoning: l = √(7² + 24²) = √625 = 25 cm. Volume = (1/3) × (22/7) × 49 × 24 = 1 232 cm³. Curved area = (22/7) × 7 × 25 = 550 cm². Total surface area would add the base, πr² = 154 cm², giving 704 cm².
Traps that cost marks
- Using the diameter as the radius.
- Giving curved surface area when total surface area is required, or the reverse.
- Converting cm² to m² by dividing by 100 instead of 10 000.
- Quoting the length ratio as the area ratio in similar shapes.