AQA GCSE Mathematics 8300: Geometry and measures – Study Guide

This guide teaches Topic 4, Geometry and measures (references G1–G25), of the AQA GCSE Mathematics (8300) specification, Version 1.0, for exams in May/June 2017 onwards and every May/June and November series after that. It covers all three parts: 3.4.1 Properties and constructions, 3.4.2 Mensuration and calculation, and 3.4.3 Vectors. Content in the specification’s “Higher content only” column is marked Higher tier only; everything else is on both tiers. Geometry can be tested on Paper 1 (non-calculator) and on Papers 2 and 3 (calculator allowed).

When you have worked through it, use the revision notes and the practice questions. The course hub is AQA GCSE Mathematics, the printable checklist lists every statement, and the free diagnostics show where your gaps are.

What this topic covers

Reference	What you must be able to do	Tier
G1, G3, G4	Use geometric terms; angle facts; properties of triangles, quadrilaterals and polygons	Both
G2	Ruler-and-compass constructions and loci	Both
G5, G6	Congruence criteria (SSS, SAS, ASA, RHS); simple proofs	Both
G7	Rotation, reflection, translation, enlargement (including fractional scale factors)	Both
G7	Enlargement with a negative scale factor	Higher tier only
G8	Combinations of transformations	Higher tier only
G9	Circle vocabulary, including tangent, arc, sector, segment	Both
G10	Prove and apply the circle theorems	Higher tier only
G11–G13	Geometry on coordinate axes; 3D solids; plans and elevations	Both
G14, G15	Units, measuring, maps, scale drawings, bearings	Both
G16–G18	Areas, perimeters, volumes, surface areas, arcs and sectors	Both
G19	Congruence and similarity; lengths in similar figures	Both
G19	Areas and volumes in similar figures	Higher tier only
G20, G21	Pythagoras and trigonometry in 2D right-angled triangles; exact values	Both
G20	Trigonometry in general triangles and in 3D	Higher tier only
G22, G23	Sine rule, cosine rule, area = ½ab sin C	Higher tier only
G24, G25	Translations as vectors; vector arithmetic	Both
G25	Vector proofs	Higher tier only

Angles and polygons (G1, G3, G4, G6)

Facts you must quote by name:

• Angles at a point add up to 360°; angles on a straight line add up to 180°.
• Vertically opposite angles are equal.
• On parallel lines, alternate angles are equal and corresponding angles are equal. The specification says colloquial terms such as “Z angles” are not acceptable, so write “alternate angles”.
• Angles in a triangle add up to 180°; base angles of an isosceles triangle are equal.
• The interior angle sum of an n-sided polygon is (n − 2) × 180°. Exterior angles of any polygon add up to 360°.

Know the properties of the square, rectangle, parallelogram, trapezium, kite and rhombus, and the names pentagon, hexagon, octagon and decagon.

Worked example. Each interior angle of a regular polygon is 162°. Find the number of sides and the interior angle sum.

Exterior angle = 180° − 162° = 18°
Number of sides = 360° ÷ 18° = 20
Interior angle sum = (20 − 2) × 180° = 3240°

Constructions and loci (G2)

With a ruler and compasses you must be able to construct: the perpendicular bisector of a line segment, a perpendicular to a line from or at a given point, the bisector of an angle, and an angle of 60° (an equilateral triangle). Leave all construction arcs visible.

Loci you should recognise:

• Fixed distance from a point: a circle.
• Equidistant from two points: the perpendicular bisector.
• Equidistant from two lines that meet: the angle bisector.

The perpendicular distance from a point to a line is the shortest distance to that line.

Congruence and transformations (G5, G7, G8, G24)

Two triangles are congruent if they satisfy SSS, SAS, ASA or RHS. In a proof, name the matching sides and angles and give the criterion.

To describe a transformation fully:

• Translation: a column vector, such as (3, −2) written vertically (3 right, 2 down).
• Reflection: the equation of the mirror line.
• Rotation: centre, angle and direction.
• Enlargement: centre and scale factor. A fractional scale factor makes the shape smaller. Higher tier only: a negative scale factor puts the image on the opposite side of the centre, upside down.

Higher tier only: describe the single transformation equivalent to a combination, and say what stays invariant (unchanged).

Circles (G9, G10)

Parts of a circle: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment.

Higher tier only: circle theorems (G10)

1. The angle at the centre is twice the angle at the circumference on the same arc.
2. The angle in a semicircle is 90°.
3. Angles in the same segment are equal.
4. Opposite angles of a cyclic quadrilateral add up to 180°.
5. A tangent is perpendicular to the radius at the point of contact.
6. Tangents from an external point are equal in length.
7. The perpendicular from the centre to a chord bisects the chord.
8. Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.

Worked example. A, B and C lie on a circle with centre O. Angle AOC = 118°, and B is on the major arc. Angle ABC = 118° ÷ 2 = 59°, because the angle at the centre is twice the angle at the circumference.

3D shapes, measures and bearings (G12–G15)

• Know the faces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres.
• A plan is the view from above; front and side elevations are views from the front and side. You must be able to interpret them and to construct them.
• Bearings are measured clockwise from north and written with three figures (e.g. 047°). The back bearing differs by 180°.
• Know the eight compass points (N, NE, E, SE, S, SW, W, NW).

Area, perimeter and volume (G16–G18)

Quantity	Formula	Status
Triangle	½ × base × height	Know
Parallelogram	base × perpendicular height	Know
Trapezium	½(a + b)h	Know or derive
Prism (including cylinder)	area of cross-section × length	Know or derive
Circle	C = 2πr = πd, A = πr²	Know
Arc length	(θ/360) × 2πr	Fraction of the circumference
Sector area	(θ/360) × πr²	Fraction of the circle’s area
Cone	V = ⅓πr²h, curved area = πrl	Given in the question
Sphere	V = (4/3)πr³, surface area = 4πr²	Given in the question

A pyramid’s volume is ⅓ × base area × perpendicular height. Answers “in terms of π” leave π as a symbol.

Worked example: sector. Radius 9 cm, angle 140°.

Arc length = 140/360 × 2 × π × 9 = 7π = 22.0 cm (3 s.f.)
Area = 140/360 × π × 9² = 31.5π = 99.0 cm² (3 s.f.)
Perimeter = 7π + 9 + 9 = 40.0 cm (3 s.f.)

Worked example: cone. Radius 5 cm, perpendicular height 12 cm.

Slant height l = √(5² + 12²) = 13 cm
Volume = ⅓ × π × 25 × 12 = 100π cm³
Curved surface area = π × 5 × 13 = 65π cm²
Total surface area = 65π + 25π = 90π = 283 cm² (3 s.f.)

Frustums (a cone with its top cut off) are composite solids: subtract the small cone from the large one.

Similarity (G19)

In similar shapes, corresponding angles are equal and lengths are in the ratio 1 : k.

Higher tier only: areas scale by k² and volumes by k³.

Worked example (Higher). Two similar bottles are 12 cm and 18 cm tall. The small one holds 400 cm³.

k = 18 ÷ 12 = 1.5
Volume factor = 1.5³ = 3.375
Large volume = 400 × 3.375 = 1350 cm³

Pythagoras and trigonometry (G20, G21)

In a right-angled triangle with hypotenuse c: a² + b² = c², sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj. You must know these; they are not given.

Worked example. Hypotenuse 11 cm, angle 38°. Opposite side = 11 × sin 38° = 6.77 cm (3 s.f.). If opposite = 4 and adjacent = 7, θ = tan⁻¹(4/7) = 29.7°.

Exact values (learn these for questions without a calculator):

θ	0°	30°	45°	60°	90°
sin θ	0	1/2	√2/2	√3/2	1
cos θ	1	√3/2	√2/2	1/2	0
tan θ	0	√3/3	1	√3	(not required)

Non-calculator example: hypotenuse 10, angle 30°. Opposite = 10 × ½ = 5; adjacent = 10 × √3/2 = 5√3.

Higher tier only: 3D. A cuboid has a 3 cm by 4 cm base and height 12 cm. Base diagonal = √(3² + 4²) = 5 cm. Space diagonal = √(5² + 12²) = 13 cm. Angle between the space diagonal and the base = tan⁻¹(12/5) = 67.4°.

Sine rule, cosine rule and area (G22, G23) – Higher tier only

• Labelling convention (G1): side a is opposite angle A, side b is opposite angle B, and so on.
• Sine rule: a/sin A = b/sin B = c/sin C. Use it with a side and its opposite angle.
• Cosine rule: a² = b² + c² − 2bc cos A. Use it with two sides and the included angle, or three sides.
• Area = ½ab sin C.

Worked example. Sides 7 cm and 9 cm with an included angle of 48°.

a² = 7² + 9² − 2 × 7 × 9 × cos 48° = 45.689…
a = 6.76 cm (3 s.f.)
Area = ½ × 7 × 9 × sin 48° = 23.4 cm² (3 s.f.)

In another triangle A = 41°, a = 8.3 cm and B = 63°. Then b = 8.3 × sin 63° ÷ sin 41° = 11.3 cm (3 s.f.).

Vectors (G24, G25)

Add and subtract column vectors component by component, and multiply by a scalar: 2(3, −1) + (−4, 5) = (2, 3). On diagrams, AB = AO + OB = −a + b.

Higher tier only: vector proof. OA = a, OB = b. M is the midpoint of AB and N is the midpoint of OA.

OM = a + ½(b − a) = ½a + ½b
NM = OM − ON = ½a + ½b − ½a = ½b

NM is a multiple of OB, so NM is parallel to OB and half its length.

Common errors

• Writing “Z angles” or “F angles” as a reason; use “alternate” and “corresponding”.
• Using the slant height as the perpendicular height in a cone volume.
• Using the diameter in πr².
• Scaling an area by k instead of k² in similar shapes.
• Measuring a bearing anticlockwise, or not from north.
• Leaving the calculator in radians.
• Using the sine rule when the cosine rule is needed (no side–opposite-angle pair known).
• Rubbing out construction arcs.

Next steps

• Condense this into the revision notes for the final weeks.
• Test yourself with the practice questions, which have fully worked answers.
• Check your coverage against the printable checklist.

Official syllabus

AQA GCSE Mathematics (8300) specification, Version 1.0 (12 September 2014), for teaching from September 2015 and exams in May/June 2017 onwards, published by AQA. Section 3.4, Geometry and measures, G1–G25.