Edexcel 4MA1Cambridge 0580
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Edexcel IGCSE Maths A (4MA1) · Foundation tier · Revision notes

IGCSE Maths 4MA1 Foundation Geometry and Trigonometry Notes

Revision notes for geometry and trigonometry at Foundation tier (Papers 1F and 2F, grades 5 to 1). Each sub-topic has what you need to know, the key methods, a common mistake and a worked example. Cover the worked solution and try the example first.

Geometry and Trigonometry questions with answers →

4.1 Angles, lines and triangles

What you need to know: types of angle; angle facts for points, straight lines and parallel lines; angles in triangles, including the exterior angle.

  • Angles on a straight line add up to 180°, angles at a point to 360°, and vertically opposite angles are equal.
  • With parallel lines, alternate angles are equal, corresponding angles are equal and co-interior (allied) angles add up to 180°.
  • The angles of a triangle add up to 180°, and the exterior angle equals the sum of the two interior opposite angles.
  • An isosceles triangle has two equal sides and two equal base angles; an equilateral triangle has three angles of 60°.

Watch out: Give the reason with each angle you work out, using the full fact, not just 'parallel lines'.

Worked example (2 marks, grade 3):

Two parallel lines are crossed by a straight line (a transversal). One of the angles it makes with the first parallel line is 63°.

(a) Write down the size of the alternate angle at the second parallel line.

(b) Work out the size of the co-interior (allied) angle to the 63° angle.

  1. (a) Alternate angles are equal: 63°.
  2. (b) Co-interior (allied) angles add up to 180°: 180 − 63 = 117°.

Answer: (a) \(63°\) (b) \(117°\)

Practise: Angles, lines and triangles questions with answers · Unit 5 practice (Foundation) · Higher level: Angles and Polygons questions

4.2 Polygons

What you need to know: names and properties of polygons and quadrilaterals; interior and exterior angles; congruent shapes.

  • The angles of a quadrilateral add up to 360°, and a polygon with n sides has an angle sum of (n − 2) × 180°.
  • The exterior angles of any polygon add up to 360°, so a regular polygon with n sides has exterior angles of 360° ÷ n.
  • Interior angle + exterior angle = 180°.
  • Know the properties of the square, rectangle, rhombus, parallelogram, trapezium and kite; congruent shapes are exactly the same shape and size.

Watch out: Find the exterior angle first: it is the quickest route to the number of sides.

Worked example (2 marks, grade 3):

Work out the size of each interior angle of a regular octagon.

  1. Each exterior angle is 360 ÷ 8 = 45°.
  2. Each interior angle is 180 − 45 = 135°. (Check: 8 × 135 = 1080 = (8 − 2) × 180.)

Answer: \(135°\)

Practise: Polygons questions with answers · Unit 5 practice (Foundation) · Higher level: Angles and Polygons questions · Quadrilaterals and Symmetry questions

4.3 Symmetry

What you need to know: lines of symmetry and order of rotational symmetry.

  • A line of symmetry divides a shape into two mirror-image halves.
  • The order of rotational symmetry is the number of times a shape looks the same in one full turn.
  • A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n.
  • A shape with no rotational symmetry has order 1.

Watch out: The diagonals of a rectangle that is not a square are not lines of symmetry.

Worked example (2 marks, grade 3):

Here are six capital letters: H, N, S, T, Z, E

Write down the letters that have rotational symmetry of order 2.

  1. H, N, S and Z look the same after a half turn.
  2. T and E only look the same after a full turn (order 1).

Answer: H, N, S, Z

Practise: Symmetry questions with answers · Unit 5 practice (Foundation) · Higher level: Quadrilaterals and Symmetry questions

4.4 Measures

What you need to know: reading scales and estimating measures; 12-hour and 24-hour times; three-figure bearings; speed, density and pressure.

  • Average speed = distance ÷ time; rearrange for distance or time. Convert minutes to hours by dividing by 60.
  • Density = mass ÷ volume and pressure = force ÷ area.
  • Bearings are measured clockwise from north and written with three figures, such as 070°.
  • A back bearing differs by 180°.

Watch out: For an average speed over two parts of a journey, use total distance ÷ total time, not the mean of the two speeds.

Worked example (2 marks, grade 3):

The bearing of \(B\) from \(A\) is 070°.

Work out the bearing of \(A\) from \(B\).

  1. The back bearing differs by 180°.
  2. 70 + 180 = 250°.

Answer: \(250°\)

Practise: Measures questions with answers · Unit 3 practice (Foundation) · Unit 5 practice (Foundation) · Higher level: Units and Compound Measures questions · Bearings and Constructions questions

4.5 Construction

What you need to know: accurate drawing with ruler, protractor and compasses; scale drawings; perpendicular bisectors and angle bisectors.

  • Measure lengths to the nearest millimetre and angles to the nearest degree.
  • The perpendicular bisector of AB cuts AB in half at right angles; every point on it is the same distance from A and from B.
  • An angle bisector cuts an angle into two equal parts.
  • On a scale drawing, multiply drawing lengths by the scale to get real lengths, and divide real lengths by the scale to get drawing lengths.

Watch out: Leave your construction arcs on the diagram: they show your method.

Worked example (2 marks, grade 3):

A scale drawing of a room uses a scale of 1 cm to 2 m. The room is 7.5 m long and 5 m wide.

Work out the length and width of the room on the drawing.

  1. Divide each real length (in m) by 2 to get centimetres on the drawing.
  2. 7.5 ÷ 2 = 3.75 cm and 5 ÷ 2 = 2.5 cm.

Answer: \(3.75\) cm by \(2.5\) cm

Practise: Construction questions with answers · Unit 5 practice (Foundation) · Higher level: Bearings and Constructions questions

4.6 Circle properties

What you need to know: the parts of a circle; tangent and chord facts.

  • Know the words: centre, radius, diameter, chord, circumference, arc, sector, segment and tangent.
  • A tangent is perpendicular to the radius at the point where it touches the circle.
  • The two tangents drawn to a circle from one outside point are equal in length.
  • The line from the centre perpendicular to a chord cuts the chord in half.

Watch out: A right angle between a tangent and a radius often means you can use Pythagoras' theorem.

Worked example (2 marks, grade 3):

\(O\) is the centre of a circle. \(PA\) is a tangent to the circle at \(A\). Angle \(OPA = 32°\).

Work out the size of angle \(AOP\). Give reasons.

  1. A tangent is perpendicular to the radius at the point of contact, so angle OAP = 90°.
  2. Angles in triangle OAP add up to 180°: 180 − 90 − 32 = 58°.

Answer: \(58°\)

Higher tier only: intersecting chords, cyclic quadrilaterals and the circle theorems.

Practise: Circle properties questions with answers · Unit 5 practice (Foundation) · Higher level: Circle Theorems questions

4.7 Geometrical reasoning

What you need to know: giving reasons for angle answers with lines, triangles and polygons.

  • Each step in an angle problem needs a reason, such as 'angles on a straight line add up to 180°'.
  • Use the exact names: alternate, corresponding and co-interior angles; base angles of an isosceles triangle; the angle sum of a polygon.
  • Write the size of each angle as you find it and say which angle it is, such as angle ABC = 72°.
  • Look for isosceles triangles: equal sides, including two radii of a circle, give equal angles.

Watch out: 'Z angles' and 'F angles' are not accepted as reasons: write alternate and corresponding.

Worked example (2 marks, grade 3):

In triangle \(ABC\), angle \(A = 48°\) and angle \(B = 67°\). The side \(BC\) is extended to a point \(D\).

Work out the size of angle \(ACD\). Give a reason.

  1. The exterior angle ACD equals angle A + angle B = 48 + 67 = 115°.
  2. (Or: angle ACB = 180 − 48 − 67 = 65°, and 180 − 65 = 115° on the straight line.)

Answer: \(115°\) (exterior angle of a triangle = sum of the interior opposite angles)

Higher tier only: reasons in any context, including circle theorems.

Practise: Geometrical reasoning questions with answers · Unit 5 practice (Foundation) · Higher level: Angles and Polygons questions · Circle Theorems questions

4.8 Trigonometry and Pythagoras' theorem

What you need to know: Pythagoras' theorem in 2D; sine, cosine and tangent in right-angled triangles; bearings problems.

  • Pythagoras' theorem: in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
  • To find a shorter side, subtract: a² = c² − b².
  • SOH CAH TOA: sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse, tan = opposite ÷ adjacent.
  • To find an angle, use the inverse function on your calculator, such as tan⁻¹(5 ÷ 8).

Watch out: Label the sides opposite, adjacent and hypotenuse from the angle you are using before you choose the ratio.

Worked example (2 marks, grade 3):

A right-angled triangle has a hypotenuse of 13 cm and one other side of 5 cm.

Work out the length of the third side.

  1. The third side is a shorter side: a² = 13² − 5² = 169 − 25 = 144.
  2. a = 12 cm.

Answer: \(12\) cm

Higher tier only: obtuse angles, elevation and depression, the sine and cosine rules, ½ab sin C and three dimensions.

Practise: Trigonometry and Pythagoras' theorem questions with answers · Unit 5 practice (Foundation) · Higher level: Pythagoras' Theorem questions · Right-Angled Trigonometry questions

4.9 Mensuration of 2D shapes

What you need to know: perimeter and area of triangles, rectangles, parallelograms and trapezia; circumference and area of circles and semicircles; area units.

  • Area of a rectangle = length × width; of a triangle = ½ × base × height; of a parallelogram = base × height.
  • Area of a trapezium = ½(a + b)h, where a and b are the parallel sides.
  • Circumference = πd = 2πr and area of a circle = πr².
  • A semicircle's perimeter is half the circumference plus the diameter.

Watch out: Use the perpendicular height, not the slanted side, in area formulae.

Worked example (2 marks, grade 3):

A circle has a diameter of 12 cm.

Work out its area. Give your answer correct to 3 significant figures.

  1. The radius is 6 cm.
  2. Area = πr² = π × 36 = 113.09… = 113 cm².

Answer: \(113\) cm\(^2\)

Higher tier only: arcs and sectors.

Practise: Mensuration of 2D shapes questions with answers · Unit 5 practice (Foundation) · Higher level: Area and Perimeter questions · Circles, Arcs and Sectors questions

4.10 3D shapes and volume

What you need to know: names and features of solids; surface area, including cylinders; volume of prisms and cylinders; volume units and litres.

  • Volume of a prism = area of cross-section × length; a cylinder is a prism with a circular cross-section, so V = πr²h.
  • Surface area is the total area of all the faces; a cylinder's curved surface opens out into a rectangle of area 2πrh.
  • A cuboid has 6 faces, 12 edges and 8 vertices.
  • 1 litre = 1000 cm³ and 1 m³ = 1 000 000 cm³.

Watch out: A closed cylinder has two circular ends: include both in its surface area.

Worked example (2 marks, grade 3):

A cuboid is 6 cm long, 4 cm wide and 2 cm high.

Work out its total surface area.

  1. The faces come in pairs: 6 × 4 = 24, 6 × 2 = 12, 4 × 2 = 8.
  2. Total = 2 × (24 + 12 + 8) = 88 cm².

Answer: \(88\) cm\(^2\)

Higher tier only: surface area and volume of spheres and cones.

Practise: 3D shapes and volume questions with answers · Unit 5 practice (Foundation) · Higher level: Volume and Surface Area questions

4.11 Similarity

What you need to know: similar shapes and scale factors for lengths; maps and scale drawings.

  • Similar shapes have the same angles, and all their corresponding lengths are in the same ratio (the scale factor).
  • Scale factor = a length on the new shape ÷ the matching length on the original shape.
  • When a line is parallel to one side of a triangle, the small triangle and the whole triangle are similar.
  • Map scales and scale drawings use the same idea: real length = drawing length × scale.

Watch out: Match the sides carefully: the scale factor must compare corresponding sides.

Worked example (3 marks, grade 3):

Triangles \(ABC\) and \(DEF\) are similar, with \(A\) matching \(D\), \(B\) matching \(E\) and \(C\) matching \(F\). \(AB = 5\) cm, \(DE = 15\) cm, \(BC = 7\) cm and angle \(A = 40°\).

(a) Work out the length of \(EF\).

(b) Write down the size of angle \(D\).

  1. (a) Scale factor = 15 ÷ 5 = 3, so EF = 7 × 3 = 21 cm.
  2. (b) Corresponding angles of similar shapes are equal: angle D = 40°.

Answer: (a) \(21\) cm (b) \(40°\)

Higher tier only: areas and volumes of similar figures.

Practise: Similarity questions with answers · Unit 5 practice (Foundation) · Higher level: Bearings and Constructions questions · Similar Shapes questions

Topic list for Edexcel IGCSE Maths 4MA1 Foundation geometry and trigonometry: angles, lines and triangles, polygons, symmetry, measures, construction, circle properties, geometrical reasoning, trigonometry and pythagoras' theorem, mensuration of 2d shapes, 3d shapes and volume, similarity
The 11 Foundation sub-topics of geometry and trigonometry.

Practise geometry and trigonometry