What you'll learn
This revision guide covers all geometry and measures content tested in WJEC GCSE Mathematics. You'll master angle properties, circle theorems, polygons, transformations, area, volume, and mensuration. These topics account for approximately 25-30% of both Foundation and Higher tier papers.
Key terms and definitions
Congruent — shapes that are identical in size and shape, though orientation may differ
Similar — shapes with the same angles but different sizes; corresponding lengths are in proportion
Chord — a straight line segment joining two points on a circle's circumference
Tangent — a straight line that touches a circle at exactly one point, perpendicular to the radius at that point
Sector — the region enclosed by two radii and an arc of a circle
Prism — a 3D shape with a constant cross-section throughout its length
Perpendicular bisector — a line that cuts another line segment exactly in half at 90°
Locus — the path or set of points satisfying a given condition
Core concepts
Angle properties and polygons
Angles on a straight line sum to 180°. Angles around a point sum to 360°.
Vertically opposite angles are equal when two straight lines intersect.
Parallel lines cut by a transversal create:
- Alternate angles (equal, Z-pattern)
- Corresponding angles (equal, F-pattern)
- Co-interior angles (sum to 180°, C-pattern)
Interior angles of an n-sided polygon sum to (n − 2) × 180°.
Each exterior angle of a regular polygon = 360° ÷ n, where n is the number of sides.
For regular polygons, each interior angle = 180° − exterior angle.
Circle theorems
Learn these nine circle theorems for Higher tier (Foundation tier requires knowledge of the first three):
- Angle at the centre is twice the angle at the circumference (both subtended by the same arc)
- Angle in a semicircle equals 90°
- Angles in the same segment are equal
- Opposite angles in a cyclic quadrilateral sum to 180°
- Tangent perpendicular to radius at the point of contact
- Two tangents from an external point are equal in length
- Alternate segment theorem: angle between tangent and chord equals angle in alternate segment
- Angle between tangent and radius is 90°
- Perpendicular from centre to chord bisects the chord
Circle arc length = (θ/360) × 2πr, where θ is the angle in degrees.
Circle sector area = (θ/360) × πr².
Circumference = 2πr or πd, where r = radius, d = diameter.
Area of circle = πr².
Transformations
Four types appear at GCSE:
Translation: sliding movement described by a column vector (x-shift / y-shift). Every point moves the same distance and direction.
Reflection: mirror image across a line of symmetry. Measure perpendicular distances from the mirror line; object and image points are equidistant.
Rotation: turning about a fixed point. Specify three elements: centre of rotation, angle, direction (clockwise/anticlockwise).
Enlargement: scaling from a centre. Specify centre of enlargement and scale factor. Scale factor k < 1 creates reduction; k > 1 creates enlargement; negative k produces enlargement on opposite side of centre.
Area scale factor = (length scale factor)²
Volume scale factor = (length scale factor)³
Congruent transformations preserve size and shape: translation, reflection, rotation.
Similar transformations preserve shape only: enlargement.
Area and perimeter
Rectangle: area = length × width; perimeter = 2(length + width)
Triangle: area = ½ × base × perpendicular height
Parallelogram: area = base × perpendicular height
Trapezium: area = ½(a + b)h, where a and b are parallel sides, h is perpendicular distance between them
Circle: area = πr²; circumference = πd = 2πr
Compound shapes: split into recognizable components, calculate each area, add or subtract as appropriate.
For irregular shapes on grids, count whole squares and estimate part-squares.
Volume and surface area
Cube: volume = side³; surface area = 6 × side²
Cuboid: volume = length × width × height; surface area = 2(lw + lh + wh)
Prism: volume = area of cross-section × length
Cylinder: volume = πr²h; curved surface area = 2πrh; total surface area = 2πrh + 2πr²
Pyramid: volume = ⅓ × base area × perpendicular height
Cone: volume = ⅓πr²h; curved surface area = πrl (where l = slant height)
Sphere: volume = ⁴⁄₃πr³; surface area = 4πr²
For composite solids, break down into standard shapes and add/subtract volumes appropriately.
Pythagoras' theorem and trigonometry
Pythagoras' theorem: In a right-angled triangle, a² + b² = c², where c is the hypotenuse.
Use to find:
- Hypotenuse: c = √(a² + b²)
- Shorter side: a = √(c² − b²)
Works in 2D and 3D problems (find diagonal in cuboids by applying theorem twice).
Trigonometric ratios for right-angled triangles:
- sin θ = opposite/hypotenuse
- cos θ = adjacent/hypotenuse
- tan θ = opposite/adjacent
To find sides: rearrange to give side = hypotenuse × sin θ (etc.)
To find angles: use inverse functions (sin⁻¹, cos⁻¹, tan⁻¹)
Sine rule: a/sin A = b/sin B = c/sin C (use to find sides or angles in non-right-angled triangles)
Cosine rule: a² = b² + c² − 2bc cos A (use when given two sides and included angle, or three sides)
Area of triangle = ½ab sin C (use when given two sides and included angle)
Bearings are three-figure angles measured clockwise from north (000° to 360°).
Construction and loci
Standard constructions using ruler and compasses only:
- Perpendicular bisector of a line segment: arcs from both endpoints, same radius > half the length
- Angle bisector: arcs from vertex, then arcs from intersections on arms
- Perpendicular from/to a line: arcs centred on line points or from external point
- 60° angle: radius from point on line creates equilateral triangle
Common loci:
- Points equidistant from a point: circle
- Points equidistant from a line: parallel lines either side
- Points equidistant from two points: perpendicular bisector
- Points equidistant from two lines: angle bisector
- Path of an object moving under constraints: precise drawing required
Use pencil, ruler and compasses. Leave construction arcs visible for method marks.
Worked examples
Example 1: Circle theorem application (Higher tier)
Question: In the diagram, O is the centre of the circle. Points A, B and C lie on the circumference. Angle AOC = 136°. Calculate angle ABC. [2 marks]
Solution:
Angle at centre = 136°
Angle at circumference = 136° ÷ 2 [1 mark for method]
Angle ABC = 68° [1 mark for answer]
Reasoning: The angle at the centre is twice the angle at the circumference when subtended by the same arc.
Example 2: Volume of composite solid
Question: A solid shape consists of a cylinder of radius 5 cm and height 12 cm with a hemisphere of radius 5 cm on top. Calculate the total volume, giving your answer to 3 significant figures. [4 marks]
Solution:
Volume of cylinder = πr²h = π × 5² × 12 = 300π cm³ [1 mark]
Volume of hemisphere = ½ × ⁴⁄₃πr³ = ⅔π × 5³ = ⅔π × 125 = 250π/3 cm³ [1 mark]
Total volume = 300π + 250π/3 = 900π/3 + 250π/3 = 1150π/3 [1 mark for combining]
= 1206.4 cm³ (3 s.f.) [1 mark for final answer]
Example 3: Trigonometry in context
Question: A ladder of length 6.5 m leans against a wall. The foot of the ladder is 2.3 m from the base of the wall. Calculate the angle the ladder makes with the ground. [3 marks]
Solution:
Draw right-angled triangle: hypotenuse = 6.5 m, adjacent = 2.3 m [1 mark for diagram/identification]
cos θ = adjacent/hypotenuse = 2.3/6.5 [1 mark for correct ratio]
θ = cos⁻¹(2.3/6.5) = 69.3° (1 d.p.) [1 mark for answer]
Common mistakes and how to avoid them
Mixing up arc length and sector area formulae — remember arc length is a distance (linear), sector area is 2D. Check units in the question.
Forgetting to halve when using "angle at centre is twice angle at circumference" — always identify which angle is at the centre and which is at the circumference first.
Using degrees instead of the correct trigonometric ratio — SOH CAH TOA helps identify which ratio to use; check your calculator is in degree mode.
Not showing construction arcs — construction questions award method marks for visible arcs; don't erase them.
Confusing similar and congruent — congruent means identical (same size and shape); similar means same shape, different size.
Applying Pythagoras to non-right-angled triangles — check for the right angle marker; use sine or cosine rule for other triangles.
Exam technique for Geometry and Measures
"Calculate" requires working shown and a numerical answer; "measure" means use a ruler/protractor and give answer with units; "construct" means use ruler and compasses only with construction arcs visible.
For circle theorem questions, write which theorem you're using — this can earn method marks even if your calculation is incorrect.
In transformation questions, always state all required information: for rotation (centre, angle, direction), for enlargement (centre, scale factor), for reflection (equation of mirror line).
Volume and area questions often have 3-4 marks: expect one mark for formula, one for substitution, one for working, one for final answer with correct units.
Quick revision summary
Geometry and Measures covers angles (including circle theorems at Higher), transformations, area, perimeter, volume, surface area, Pythagoras, trigonometry, and constructions. Master angle properties in parallel lines and polygons. Know all circle theorems and when to apply them. Understand the four transformations and their properties. Learn formulae for area and volume of 2D shapes and 3D solids. Apply Pythagoras and trigonometry to right-angled triangles; use sine and cosine rules for other triangles. Practice constructions with ruler and compasses, leaving arcs visible.