Geometry and Measures

What you'll learn

Geometry and Measures forms approximately 20% of your Edexcel GCSE Mathematics examination. This topic encompasses angle properties, polygons, circle theorems, transformations, area and volume calculations, and unit conversions. Mastery of these concepts is essential for both Foundation and Higher tier candidates.

Key terms and definitions

Congruent — shapes that are identical in size and shape; corresponding sides and angles are equal

Similar — shapes with the same angles but different sizes; corresponding sides are in the same ratio (scale factor)

Circumference — the perimeter of a circle, calculated using C = πd or C = 2πr

Sector — a portion of a circle enclosed by two radii and an arc, resembling a slice of pie

Invariant — a property that remains unchanged after a transformation, such as length under translation or rotation

Perpendicular bisector — a line that divides another line segment into two equal parts at 90°

Locus — the set of all points that satisfy a particular condition or rule

Pythagoras' theorem — for right-angled triangles: a² + b² = c², where c is the hypotenuse

Core concepts

Angle properties and polygons

Angle relationships:

• Angles on a straight line sum to 180°
• Angles around a point sum to 360°
• Vertically opposite angles are equal
• Corresponding angles are equal (parallel lines)
• Alternate angles are equal (parallel lines)
• Co-interior angles sum to 180° (parallel lines)

Polygon angle formulae:

• Sum of interior angles = (n - 2) × 180°, where n is the number of sides
• Each interior angle of a regular polygon = (n - 2) × 180° ÷ n
• Each exterior angle of a regular polygon = 360° ÷ n
• Exterior angles of any polygon sum to 360°

For a regular hexagon: each interior angle = (6 - 2) × 180° ÷ 6 = 120°

Circle theorems

These are Higher tier only but fundamental for top grades:

Key circle theorems:

• The angle in a semicircle is 90°
• The angle at the centre is twice the angle at the circumference (subtended by the same arc)
• Angles in the same segment are equal
• Opposite angles in a cyclic quadrilateral sum to 180°
• The angle between a tangent and a radius is 90°
• Two tangents from an external point are equal in length
• The alternate segment theorem: the angle between a tangent and chord equals the angle in the alternate segment

Circle calculations:

• Circumference = πd = 2πr
• Area of circle = πr²
• Arc length = (θ/360) × 2πr (where θ is the sector angle)
• Sector area = (θ/360) × πr²

Perimeter, area and volume

2D shapes:

• Rectangle: Area = length × width
• Triangle: Area = ½ × base × height
• Parallelogram: Area = base × perpendicular height
• Trapezium: Area = ½(a + b)h, where a and b are parallel sides
• Circle: Area = πr²

3D solids:

• 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
• Pyramid: Volume = ⅓ × base area × height
• Cone: Volume = ⅓πr²h; Curved surface area = πrl (where l is slant height)
• Sphere: Volume = 4/3πr³; Surface area = 4πr²

Compound shapes: Split complex shapes into recognisable components, calculate each separately, then add or subtract as appropriate.

Pythagoras' theorem and trigonometry

Pythagoras' theorem applies to right-angled triangles only:

• a² + b² = c², where c is the hypotenuse (longest side, opposite the right angle)
• Use to find a missing side length when two sides are known
• Can be used in 3D problems by finding intermediate lengths

Trigonometric ratios (SOHCAHTOA):

• sin θ = opposite/hypotenuse
• cos θ = adjacent/hypotenuse
• tan θ = opposite/adjacent

For finding angles: use inverse functions (sin⁻¹, cos⁻¹, tan⁻¹)

Exact trigonometric values (Higher tier):

• sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3
• sin 45° = 1/√2, cos 45° = 1/√2, tan 45° = 1
• sin 60° = √3/2, cos 60° = ½, tan 60° = √3

Sine rule and cosine rule (Higher tier):

• Sine rule: a/sin A = b/sin B = c/sin C
• Cosine rule: a² = b² + c² - 2bc cos A
• Area of triangle = ½ab sin C

Transformations

Four transformation types:

Translation: sliding a shape without rotation

• Described by a column vector: (x/y) where x is horizontal movement, y is vertical movement
• Positive x is right, positive y is up
• All points move by the same vector

Rotation: turning about a fixed point

• Requires: centre of rotation, angle (direction matters: clockwise or anticlockwise), direction
• Use tracing paper in exams to check your rotation

Reflection: flipping over a mirror line

• Requires: equation of the mirror line (e.g., x = 2, y = x, y = -x)
• Each point and its image are equidistant from the mirror line

Enlargement: changing size (possibly position)

• Requires: centre of enlargement, scale factor
• Scale factor > 1: shape gets larger
• Scale factor between 0 and 1: shape gets smaller
• Negative scale factor: shape inverts through the centre
• Area scale factor = (length scale factor)²

Congruence and similarity

Congruence conditions: Four ways to prove triangles are congruent:

• SSS (three sides equal)
• SAS (two sides and the included angle equal)
• ASA (two angles and a corresponding side equal)
• RHS (right angle, hypotenuse, and one other side equal)

Similarity: Shapes are similar when:

• Corresponding angles are equal
• Corresponding sides are in the same ratio

Scale factor applications:

• If linear scale factor = k, then area scale factor = k²
• If linear scale factor = k, then volume scale factor = k³
• Use these relationships to solve problems involving similar shapes

Constructions and loci

Standard constructions (using ruler and compasses only):

• Perpendicular bisector of a line segment
• Bisector of an angle
• Perpendicular from a point to a line
• Perpendicular from a point on a line

Common loci:

• Points equidistant from a fixed point: circle
• Points equidistant from two points: perpendicular bisector
• Points equidistant from a line: two parallel lines
• Points equidistant from two lines: angle bisector

Regions satisfying multiple conditions: Combine loci by identifying the intersection region. Shade or label clearly according to the question instruction.

Units and measures

Length conversions:

• 1 km = 1000 m
• 1 m = 100 cm
• 1 cm = 10 mm
• 1 mile ≈ 1.6 km; 1 inch = 2.54 cm

Area conversions:

• 1 m² = 10,000 cm² (not 100)
• 1 cm² = 100 mm²
• 1 hectare = 10,000 m²

Volume conversions:

• 1 m³ = 1,000,000 cm³
• 1 litre = 1000 cm³
• 1 ml = 1 cm³

Compound measures:

• Speed = distance ÷ time
• Density = mass ÷ volume
• Pressure = force ÷ area

Worked examples

Example 1: Circle theorem application (Higher)

Question: In the diagram, O is the centre of the circle. Angle BAC = 35°. Calculate angle BOC. Give a reason for your answer. [2 marks]

Solution: Angle BOC = 2 × 35° = 70° [1 mark]

Reason: The angle at the centre is twice the angle at the circumference [1 mark]

(Alternative wording accepted: "Angle at centre = 2 × angle at circumference" or similar clear statement of the theorem)

Example 2: Volume of compound solid

Question: A toy consists of a cylinder with radius 4 cm and height 10 cm, topped with a hemisphere of the same radius. Calculate the total volume of the toy. Give your answer correct to 3 significant figures. [4 marks]

Solution: Volume of cylinder = πr²h = π × 4² × 10 = 160π cm³ [1 mark]

Volume of hemisphere = ½ × 4/3πr³ = ⅔ × π × 4³ = ⅔ × π × 64 = 128π/3 cm³ [1 mark]

Total volume = 160π + 128π/3 = 480π/3 + 128π/3 = 608π/3 [1 mark]

= 636.17... = 636 cm³ (3 s.f.) [1 mark]

Example 3: Similar shapes

Question: Two similar cylinders have heights of 6 cm and 9 cm. The smaller cylinder has a volume of 120 cm³. Calculate the volume of the larger cylinder. [3 marks]

Solution: Linear scale factor = 9 ÷ 6 = 1.5 [1 mark]

Volume scale factor = 1.5³ = 3.375 [1 mark]

Volume of larger cylinder = 120 × 3.375 = 405 cm³ [1 mark]

Common mistakes and how to avoid them

• Confusing diameter and radius: Always identify which measurement you have been given. Remember radius = diameter ÷ 2. Many circle problems give the diameter but formulae use radius.

• Incorrect angle reasoning: When proving angle facts, you must state the geometrical reason (e.g., "angles on a straight line"). Simply calculating the answer without justification loses marks.

• Area vs perimeter confusion: Read the question carefully. Area is measured in square units (cm², m²); perimeter in linear units (cm, m). Using the wrong formula loses all marks.

• Scale factor errors with area and volume: Remember that if the linear scale factor is k, area scales by k² and volume by k³. Students often multiply area by k instead of k².

• Forgetting to convert units: When calculating with mixed units (e.g., metres and centimetres), convert everything to the same unit first. Check your final answer makes sense.

• Rounding too early: Keep full calculator values until the final answer. Rounding intermediate steps introduces errors and loses accuracy marks. Use the ANS button effectively.

Exam technique for Geometry and Measures

• "Calculate" requires working: Show your method clearly. Writing just an answer, even if correct, may only earn the final mark. Show formulae used and substitution of values.

• "Show that" questions require precision: You must demonstrate every step leading to the given answer. These typically test whether you can apply formulae and theorems correctly. Full marks require reaching the exact value stated.

• Draw diagrams when not provided: For circle theorem, Pythagoras, or trigonometry problems, sketch and label a diagram. This reduces errors and helps you identify the correct approach.

• Check units in your answer: Questions often specify required units. Converting m² to cm² at the end requires multiplying by 10,000 (not 100). Write units with your final answer unless the question specifies otherwise.

Quick revision summary

Geometry and Measures covers angle properties in polygons and with parallel lines, circle theorems (Higher), area and volume calculations for 2D and 3D shapes, Pythagoras and trigonometry, the four transformations, congruence and similarity, constructions and loci, and unit conversions. Remember to show full working, state geometrical reasons when required, and apply correct scale factors (k for length, k² for area, k³ for volume). Always check your units match the question requirements.