Shape, Space and Measures
This domain tests geometrical reasoning, accurate measurement and mensuration. SEC questions ask you to calculate angles with reasons, use Pythagoras and right-angled trigonometry, find lengths, areas and volumes, work with similar shapes and describe transformations. Diagrams are usually not drawn to scale, so do not measure angles with a protractor unless told to.
Angle and circle facts
- Angles on a straight line sum to 180°; angles at a point sum to 360°.
- Vertically opposite angles are equal; alternate and corresponding angles arise between parallel lines.
- Interior angle sum of an n-sided polygon is (n − 2) × 180°; each exterior angle of a regular polygon is 360°/n.
- Angle in a semicircle is 90°; a tangent is perpendicular to the radius at the point of contact; opposite angles of a cyclic quadrilateral sum to 180°.
- Bearings are measured clockwise from north and written with three digits.
Pythagoras, trigonometry and measurement
- Pythagoras: a² + b² = c², where c is the hypotenuse.
- SOHCAHTOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Check the calculator is in degrees.
- Volume of a prism = area of cross-section × length; cylinder V = πr²h; cone V = (1/3)πr²h; sphere V = (4/3)πr³.
- Similar shapes: lengths scale by k, areas by k², volumes by k³.
Worked example
A ladder 6 m long leans against a wall, making 65° with the ground. How high up the wall does it reach? The height is opposite the 65° angle and 6 m is the hypotenuse, so sin 65° = h/6. Therefore h = 6 sin 65° = 6 × 0.9063 = 5.44 m (3 s.f.).
Traps
- Using the wrong trigonometric ratio or misidentifying the adjacent side.
- Mixing units, such as cm with m, in area or volume questions.
- Giving the slant height when the perpendicular height is required.
- Omitting the reason when a question says “give a reason for your answer”.