Geometry and Measures
Overview
Geometry and Measures tests angle reasoning, properties of shapes, mensuration, Pythagoras' theorem and trigonometry, transformations, constructions/loci and vectors. CCEA questions often combine topics, and a correct numerical answer without a stated angle reason may lose a mark on a “give reasons” question.
Core rules
- Angles on a straight line sum to 180°; angles at a point sum to 360°; vertically opposite angles are equal.
- The interior angle sum of an n-sided polygon is (n − 2) × 180°; the exterior angles sum to 360°.
- Pythagoras: a² + b² = c², where c is the hypotenuse. It applies only to right-angled triangles.
- SOHCAHTOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
- Similar shapes: length scale factor k, area scale factor k², volume scale factor k³.
- Bearings are measured clockwise from north and written with three figures.
- A vector has magnitude and direction; column vectors add component-wise.
Worked example
A 5 m ladder leans against a vertical wall, with its foot on horizontal ground. The top reaches 4.2 m up the wall. Find the angle the ladder makes with the ground. The side opposite the required angle is 4.2 m and the hypotenuse is 5 m, so sin θ = 4.2/5 = 0.84. θ = sin⁻¹(0.84) = 57.1° (to 1 d.p.). The angle is between the ladder and the ground, so the opposite side is the height on the wall, not the distance from the wall.
Traps
Using Pythagoras in a non-right-angled triangle; choosing the wrong trigonometric ratio; using the area scale factor for volume; measuring a bearing anticlockwise; not giving an angle reason; rounding an intermediate trig value too early.