AQA GCSE Mathematics 8300: Geometry and measures – Revision Notes

These notes cover 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. Content from the specification’s “Higher content only” column is marked Higher tier only; everything else can be set on both tiers. Geometry appears on the non-calculator Paper 1 as well as the calculator Papers 2 and 3.

For full explanations and worked examples, go back to the geometry and measures study guide. When you are ready, try the practice questions. The course hub is AQA GCSE Mathematics, the printable checklist lists every statement, and the free diagnostics help you find weak areas.

Formulas: learn, derive or given?

Formula	Status in the specification
Circumference = 2πr = πd; area of circle = πr²	Learn (not given)
Pythagoras a² + b² = c²	Learn (not given)
sin, cos, tan ratios	Learn (not given)
Sine rule, cosine rule, area = ½ab sin C	Learn (not given); Higher tier only in use
Trapezium = ½(a + b)h; prism = cross-section × length	Learn or be able to derive
Cone: V = ⅓πr²h, curved surface = πrl	Given in the question
Sphere: V = (4/3)πr³, surface area = 4πr²	Given in the question

Angles and shapes (G1, G3–G6)

• Point: 360°. Straight line: 180°. Vertically opposite: equal.
• Parallel lines: alternate angles equal, corresponding angles equal. Do not write “Z angles” or “F angles”.
• Triangle: 180°. Quadrilateral: 360°. n-gon: (n − 2) × 180°.
• Regular polygon: exterior = 360° ÷ n; interior = 180° − exterior.
• Congruence criteria: SSS, SAS, ASA, RHS. “AAA” proves similarity, not congruence.

Method in steps: angle chase with reasons

1. Mark every angle you find on the diagram.
2. After each calculation, write the fact that justifies it, in words.
3. Check that your final angle fits the diagram (acute or obtuse).

Constructions, loci and bearings (G2, G15)

Locus	Shape
Fixed distance from a point	Circle
Fixed distance from a line segment	Two parallel lines joined by semicircles
Equidistant from two points	Perpendicular bisector
Equidistant from two intersecting lines	Angle bisector

Bearings: measure from north, clockwise, three figures. Back bearing = bearing ± 180°.

Transformations (G7, G8, G24)

• Translation → column vector. Reflection → mirror line. Rotation → centre, angle, direction. Enlargement → centre, scale factor.
• Scale factor between 0 and 1 makes the image smaller.
• Higher tier only: scale factor −k: image is k times the size, on the other side of the centre, inverted.
• Higher tier only: combinations of rotations, reflections and translations; say what is invariant.

Circles (G9, G10)

Sector = “pizza slice” between two radii. Segment = region between a chord and an arc.

Higher tier only: the eight theorems to quote

Theorem	Short wording for a reason
Centre angle	Angle at the centre is twice the angle at the circumference
Semicircle	Angle in a semicircle is 90°
Same segment	Angles in the same segment are equal
Cyclic quadrilateral	Opposite angles add up to 180°
Tangent–radius	Tangent is perpendicular to the radius
Two tangents	Tangents from an external point are equal
Chord bisector	Perpendicular from the centre bisects the chord
Alternate segment	Angle between tangent and chord equals angle in the alternate segment

Mensuration (G14, G16–G18)

• Area units are squared, volume units cubed. 1 cm³ = 1 ml; 1000 cm³ = 1 litre.
• Arc length = (θ/360) × 2πr. Sector area = (θ/360) × πr².
• Sector perimeter = arc length + 2r.
• Composite solids and frustums: add or subtract simpler solids.

Worked reminder: radius 12 cm, angle 75°. Sector area = 75/360 × 144π = 30π cm²; arc length = 75/360 × 24π = 5π cm.

Similarity (G19, R12)

• Length scale factor k = new length ÷ matching old length.
• Higher tier only: area factor k², volume factor k³.
• To go backwards from areas to lengths, square-root; from volumes to lengths, cube-root.

Trigonometry (G20–G23)

Method in steps: right-angled triangle

1. Label the sides opp, adj, hyp from the angle you are using.
2. Pick the ratio with the two sides involved.
3. For a side, rearrange; for an angle, use sin⁻¹, cos⁻¹ or tan⁻¹.

Worked reminder: opposite 5, hypotenuse 8. sin x = 5/8, so x = sin⁻¹(0.625) = 38.7°.

Exact values: sin 30° = cos 60° = 1/2; sin 60° = cos 30° = √3/2; sin 45° = cos 45° = √2/2; tan 30° = √3/3; tan 45° = 1; tan 60° = √3.

Method in steps: choosing a rule (Higher tier only)

1. A side and its opposite angle known? Sine rule.
2. Two sides and the angle between them, or all three sides? Cosine rule.
3. Need an area with two sides and the included angle? ½ab sin C.
4. 3D: find a right-angled triangle inside the solid; often you need a diagonal of the base first.

Vectors (G24, G25)

• Column vectors: add and subtract each row; a scalar multiplies each row.
• Route rule: AB = AO + OB = −OA + OB.
• Higher tier only: to prove parallel, show one vector is a multiple of the other. To prove three points are collinear, show two vectors are parallel and share a point.

Small worked reminders

• Solids (G12): a triangular prism has 5 faces, 9 edges and 6 vertices. A square-based pyramid has 5 faces, 8 edges and 5 vertices.
• Plans and elevations (G13): a cylinder standing on its base has a circle as its plan and a rectangle as its front elevation.
• Parallelogram (G16): base 9 cm, perpendicular height 4 cm, slanted side 5 cm. Area = 9 × 4 = 36 cm². The 5 cm side is not used.
• Congruence (G5): two triangles with two pairs of equal sides and the angle between them equal are congruent by SAS. If the equal angle is not between the sides, SAS does not apply.
• Loci (G2): “within 3 cm of P and nearer to line AB than to line AC” means the inside of a circle of radius 3 cm centred on P, on the AB side of the bisector of angle BAC.
• Coordinates (G11): the midpoint of (2, 7) and (8, −1) is (5, 3). Use it to find centres of circles or to check that diagonals bisect each other.

Paper 1 checklist (non-calculator)

Geometry on the non-calculator paper often tests:

• angle chasing with written reasons;
• exact trigonometric values for 0°, 30°, 45°, 60° and 90°;
• Pythagoras answers left as surds, such as √45 = 3√5;
• areas, arcs and volumes “in terms of π”;
• column vector arithmetic and vector proofs;
• constructions, loci and bearings with a ruler, compasses and protractor.

Practise these without a calculator so that your arithmetic is fast and accurate.

Must-know distinctions

• Congruent vs similar: same shape and size vs same shape, different size.
• Sector vs segment: bounded by two radii vs bounded by a chord.
• Perpendicular height vs slant height: volume of a cone uses h; curved surface uses l.
• Arc length vs sector perimeter: the perimeter adds two radii.
• Plan vs elevation: from above vs from the front or side.
• Parallel vs collinear: collinear also needs a common point.

Quick self-test

1. Find the exterior angle of a regular decagon.
2. Find the interior angle sum of a heptagon.
3. The bearing of B from A is 065°. Find the bearing of A from B.
4. A trapezium has parallel sides 7 cm and 11 cm, 5 cm apart. Find its area.
5. Find the circumference of a circle of diameter 14 cm, in terms of π.
6. Write down the exact value of cos 30°.
7. Higher tier only. Two similar shapes have lengths in the ratio 2 : 5. Find the ratio of their areas.
8. Work out 3(2, −1) − (4, 5), as column vectors.
9. Find the arc length of a sector with radius 6 cm and angle 60°, in terms of π.
10. A right-angled triangle has shorter sides 9 cm and 12 cm. Find the hypotenuse.
11. Higher tier only. You know two sides and the angle between them. Which rule finds the third side?
12. A sphere has radius 3 cm. Using V = (4/3)πr³, find its volume in terms of π.

Answers

1. 360° ÷ 10 = 36°
2. (7 − 2) × 180° = 900°
3. 065° + 180° = 245°
4. ½ × (7 + 11) × 5 = 45 cm²
5. 14π cm
6. √3/2
7. 4 : 25
8. (6 − 4, −3 − 5) = (2, −8)
9. 60/360 × 12π = 2π cm
10. √(81 + 144) = √225 = 15 cm
11. The cosine rule
12. (4/3) × π × 27 = 36π cm³

Where marks are usually lost

• Giving an angle with no reason, or with a colloquial reason, when the question says “give reasons”.
• Using the diameter in πr², or the radius in πd.
• Forgetting to add the two radii to an arc length for a sector perimeter.
• Using slant height in the cone volume formula.
• Scaling area by k or volume by k² in similar figures (Higher tier only).
• Rounding the base diagonal before using it in a 3D trigonometry step.
• Calculator in radians mode, giving impossible angles.
• Choosing the sine rule without a known side–opposite-angle pair.
• Rubbing out construction arcs; they are the evidence of method.
• Proving collinearity with “parallel” only, without mentioning the shared point.

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.