Geometry and measures
Choose your practice ↓Geometry and measures is 15% of the Foundation paper and 20% of Higher, and it is the area that most rewards drawing the diagram properly. It covers geometric notation and ruler-and-compass constructions, angles at a point and on parallel lines, the angle sum of a triangle and of any polygon, the properties of triangles and of the special quadrilaterals — square, rectangle, parallelogram, trapezium, kite and rhombus — and the congruence criteria SSS, SAS, ASA and RHS. Measurement follows: area of triangles, parallelograms and trapezia, volume of prisms and cylinders, the circumference and area of a circle, arcs and sectors, surface area and volume of spheres, cones and pyramids, plans and elevations, and bearings. Pythagoras' theorem and the three trigonometric ratios sit here too, with the exact values of sin, cos and tan for the standard angles. Higher adds circle theorems and their proofs, combinations of transformations and invariance, the sine rule, the cosine rule, area = ½ab sin C, trigonometry in three dimensions, and the use of vectors to construct a geometric proof.
📚 What you learn here
- Geometric notation, drawing from a written description, and ruler-and-compass constructions and loci
- Angles at a point, on a straight line, vertically opposite, and alternate and corresponding angles on parallel lines
- The angle sum of a triangle and of any polygon, and the properties of regular polygons
- Properties of triangles and of the quadrilaterals: square, rectangle, parallelogram, trapezium, kite and rhombus
- Congruence (SSS, SAS, ASA, RHS) and similarity, including area and volume scale factors (Higher)
- Rotation, reflection, translation and enlargement, including fractional and (Higher) negative scale factors
- Area of triangles, parallelograms and trapezia; volume of cuboids, prisms and cylinders
- Circumference and area of a circle, arc length and sector area, and the surface area and volume of spheres, cones and pyramids
- Plans and elevations, bearings, scale drawings and standard units of measure
- Pythagoras' theorem, the trigonometric ratios, and the exact values of sin, cos and tan at 0°, 30°, 45°, 60° and 90°
- Circle theorems (Higher), the sine rule and cosine rule (Higher), area = ½ab sin C (Higher)
- Translations as column vectors, vector addition and scalar multiples, and vector proof (Higher)
Frequently asked questions
What is the difference between the area of a circle and its circumference?
Area is πr² and is measured in square units; circumference is 2πr, or πd, and is measured in ordinary length units. Using the diameter where the formula wants the radius is the error that produces an answer four times too big for area and twice too big for circumference.
When do I use the sine rule and when the cosine rule?
The sine rule pairs a side with the angle opposite it, so use it when you have such a pair. Use the cosine rule when you have two sides and the angle between them (SAS), or all three sides and want an angle. Both are Higher only; on a right-angled triangle use Pythagoras and the three ratios instead.
Are circle theorems on Foundation?
No. Foundation needs the parts of a circle — centre, radius, chord, diameter, circumference, tangent, arc, sector and segment — and the circle formulae, but the theorems about angles, radii, tangents and chords are Higher only.
How do I find the area of a trapezium?
Add the two parallel sides, halve the total, and multiply by the perpendicular height: ½(a + b)h. The height must be the perpendicular distance between the parallel sides, not the slanted side, which is the length the diagram usually offers you.
How is a bearing written?
As three figures, measured clockwise from north: 045°, not 45°, and 007°, not 7°. Marks are lost every year to two-figure bearings.
✍️ Written by the MathsUK teamChecked against the National Curriculum and GCSE specifications