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Edexcel IGCSE Maths (4MA1) · Revision notes · Spatial Reasoning

IGCSE Maths Geometry and Trigonometry Revision Notes

Short, plain-English notes on geometry, measurement & trigonometry for Pearson Edexcel International GCSE Mathematics A (4MA1), Higher tier. Each of the 14 sub-topics has what you need to know and one example with its answer. Cover the answer and try the example first, then practise the topic with worked solutions.

Compound Measures

Calculations with speed, distance, time, mass, density and pressure.

Compound measures combine two quantities: e.g. speed = distance ÷ time, density = mass ÷ volume, pressure = force ÷ area.

Example: A cyclist travels at 18 km/h for 1 h 45 min. Distance?

Answer: 1.75 × 18 = 31.5 km.

Practise it: Units and Compound Measures questions and answers

Area & Volume Conversions

Converting units of length, area (cm² to m²) and volume (cm³ to m³).

When converting units of area, square the linear factor (1 m² = 100² cm²). For volume, cube it.

Example: Convert 3 m² to cm².

Answer: 3 × 100² = 30 000 cm².

Practise it: Units and Compound Measures questions and answers

Geometry of Shapes

Applying angle sums, identifying parallel line transversals and quadrilaterals.

Angle rules: angles on a straight line = 180°, in a triangle = 180°, corresponding and alternate angles are equal on parallel lines.

Example: One angle of an isosceles triangle is 40°. Find the other two angles (two cases).

Answer: If 40° is the apex: 70° and 70°. If 40° is a base angle: 40° and 100°.

Practise it: Angles and Polygons questions and answers · Quadrilaterals and Symmetry questions and answers · Congruent Triangles questions and answers

Scale Drawings & Bearings

Measuring three-figure navigation bearings, map scaling.

Bearings are 3-figure angles measured clockwise from North. Back-bearing = original ± 180°.

Example: A ship sails on bearing 135°. What bearing to return?

Answer: Back-bearing = 135° + 180° = 315°.

Practise it: Bearings and Constructions questions and answers

Geometric Constructions

Constructing perpendicular bisectors, angle bisectors and standard shapes.

Constructions use compass + straight edge only: perpendicular bisector, angle bisector, 60°/90° angles.

Example: Sketch the steps for the perpendicular bisector of AB.

Answer: Arc from A above/below; same-width arc from B; connect the two intersections.

Practise it: Bearings and Constructions questions and answers

2D Perimeter & Area

Calculating composite area of triangles, parallelograms and trapezia.

Composite areas break a shape into rectangles + triangles + circles then sum.

Example: Area of a trapezium with parallel sides 12 & 18 cm, height 8 cm.

Answer: = ½(12 + 18) × 8 = 120 cm².

Practise it: Area and Perimeter questions and answers · Circles, Arcs and Sectors questions and answers

Volume of Prisms

Finding the volume of cylinders, cuboids and compound prisms.

Volume of a prism = cross-section area × length. Cylinders use πr² for the area.

Example: Triangular prism, cross-section area 24 cm², length 15 cm. Volume?

Answer: = 24 × 15 = 360 cm³.

Practise it: Volume and Surface Area questions and answers

Surface Area of Prisms

Finding composite surface area of simple and compound 3D shapes.

Surface area sums the areas of every face of a 3D solid.

Example: Cube of side 5 cm — surface area?

Answer: 6 × 5² = 150 cm².

Practise it: Volume and Surface Area questions and answers

Angles in Polygons

Calculating interior and exterior angle sums of regular / irregular polygons.

Sum of interior angles of an n-sided polygon = (n − 2) × 180°.

Example: Sum of interior angles of a regular hexagon.

Answer: (6 − 2) × 180 = 720°.

Practise it: Angles and Polygons questions and answers

Pythagoras' Theorem

Calculating lengths of missing sides in 2D right-angled triangles.

Pythagoras: in a right-angled triangle, a² + b² = c² (c = hypotenuse).

Example: Right-triangle legs 7 cm and 24 cm — hypotenuse?

Answer: √(49 + 576) = √625 = 25 cm.

Practise it: Pythagoras' Theorem questions and answers

Right-Angled Trig (SOH CAH TOA)

Calculating sides and angles, real-world application.

SOH CAH TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj — used in right-angled triangles.

Example: Hypotenuse 14 cm, angle 30°. Opposite side?

Answer: Opp = 14 × sin 30° = 7 cm.

Practise it: Right-Angled Trigonometry questions and answers

Sine & Cosine Rules

Sine rule, cosine rule and area = ½ab sin C in non-right-angled triangles.

For triangles without a right angle: sine rule a/sin A = b/sin B, cosine rule a² = b² + c² − 2bc cos A, area = ½ab sin C.

Example: Find the area of a triangle with sides 7 cm and 9 cm and included angle 40°.

Answer: ½ × 7 × 9 × sin 40° = 20.2 cm² (3 s.f.).

Practise it: Sine and Cosine Rules questions and answers

Circle Theorems

Angles at the centre and circumference, cyclic quadrilaterals, tangents and radii.

Key circle theorems: angle at the centre is twice the angle at the circumference; opposite angles of a cyclic quadrilateral add to 180°; a tangent meets a radius at 90°.

Example: The angle at the circumference is 35°. Find the angle at the centre on the same arc.

Answer: 2 × 35° = 70°.

Practise it: Circle Theorems questions and answers

Vectors

Column vectors, magnitude, and vector geometry proofs (parallel lines, collinear points).

A vector has size and direction. Add column vectors component by component; parallel vectors are multiples of each other.

Example: a = (3, −1), b = (1, 2). Find 2a + b.

Answer: (6, −2) + (1, 2) = (7, 0).

Practise it: Vectors questions and answers

More geometry and trigonometry questions

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