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Cambridge IGCSE Maths (0580) · Revision notes · Topic 4 of 9

Cambridge IGCSE Maths 0580 Geometry Notes

Revision notes for topic 4, Geometry, of the Cambridge IGCSE Mathematics (0580) syllabus for exams in 2025, 2026 and 2027. Each of the 8 sub-topics lists what the syllabus asks, marked Core or Extended, with a short explanation and a worked example. Cover the answer and try each example first.

Core Core students (Papers 1 and 3) learn the Core statements. Extended Extended students (Papers 2 and 4) learn everything: the Core statements, the Extended additions and the Extended-only sub-topics. Papers 1 and 2 are non-calculator.

4.1 Geometrical terms

Core C4.1 Extended E4.1

  • Vocabulary of points, lines, angles, parallel and perpendicular lines, bearings, similarity and congruence
  • Names and properties of triangles, quadrilaterals, polygons, circles (chord, tangent, arc, sector, segment) and solids

Extended also: The term hemisphere (not required at Core).

Partly in Edexcel 4MA1: Solid names and the vocabulary of faces, edges and vertices are not practised in 4MA1. (Congruence is vocabulary only on 0580: candidates are not asked to show shapes are congruent.)

Notes for 0580

Know the words: acute angles are less than 90°, obtuse angles between 90° and 180°, reflex angles more than 180°. Parallel lines never meet; perpendicular lines meet at 90°. Similar shapes have the same shape with sides in the same ratio; congruent shapes are identical in shape and size.

A solid has faces (flat surfaces), edges (where two faces meet) and vertices (corners). A prism has the same cross-section all the way through; a pyramid rises from its base to a single point, the apex. For these solids, faces + vertices − edges = 2, which is a quick check on a count.

Example (Core, non-calculator): How many faces, edges and vertices does (a) a triangular prism and (b) a square-based pyramid have?

  1. (a) 2 triangles + 3 rectangles = 5 faces; 3 edges on each triangle + 3 joining them = 9 edges; 6 vertices. Check: 5 + 6 − 9 = 2.
  2. (b) 1 square + 4 triangles = 5 faces; 4 base edges + 4 slant edges = 8 edges; 4 base corners + the apex = 5 vertices. Check: 5 + 5 − 8 = 2.

Answer: (a) 5 faces, 9 edges, 6 vertices (b) 5 faces, 8 edges, 5 vertices

0580 practice questions

12 questions written for 0580 in the Geometry practice: 12 Core, all non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

A cuboid is a solid. Write down the number of

(a) faces (b) edges (c) vertices.

Show the answer and mark scheme
  • B2 6 faces, 12 edges, 8 vertices (B1 for two correct)

Answer: 6 faces, 12 edges, 8 vertices

Practice question (Core, non-calculator, 3 marks):

Two identical square-based pyramids are joined together by their square bases. Write down the number of faces, edges and vertices of the new solid.

Show the answer and mark scheme
  • B1 8 faces
  • B1 12 edges
  • B1 6 vertices

Answer: 8 faces, 12 edges, 6 vertices

Edexcel 4MA1 notes: Geometry of Shapes
Practise the shared part (4MA1 questions): Angles and Polygons questions · Quadrilaterals and Symmetry questions · Congruent Triangles questions

4.2 Geometrical constructions

Core C4.2 Extended E4.2

  • Measure and draw lines and angles
  • Construct a triangle from three sides with a ruler and compasses only, showing the construction arcs
  • Draw, use and interpret nets
  • Perpendicular and angle bisector constructions and loci are not required

Partly in Edexcel 4MA1: 4MA1 constructs bisectors and loci; nets are not practised.

Notes for 0580

A net is a flat shape that folds up to make a solid. A cuboid's net is 6 rectangles; a cylinder's net is two circles and a rectangle whose length is the circumference of the circle. The area of a net is the surface area of the solid.

To construct a triangle from its three sides: draw one side with a ruler; set the compasses to the second length and draw an arc from one end; set them to the third length and draw an arc from the other end; join the point where the arcs cross to both ends. Leave the construction arcs showing.

Example (Core, non-calculator): A cuboid measures 3 cm by 4 cm by 5 cm. Find the area of its net.

  1. The net has two of each rectangle: 3 × 4, 3 × 5 and 4 × 5.
  2. 2 × (12 + 15 + 20) = 2 × 47

Answer: 94 cm²

Example (Core, non-calculator): Can a triangle be constructed with sides 4 cm, 5 cm and 10 cm? Explain.

  1. The two shorter sides add to 4 + 5 = 9 cm, which is less than 10 cm.
  2. The arcs drawn from the ends of the 10 cm side would not meet.

Answer: No: 4 + 5 < 10, so the two shorter sides cannot reach each other

0580 practice questions

12 questions written for 0580 in the Geometry practice: 12 Core, all non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 1 mark):

A net is made from 6 identical squares. Write down the name of the solid it folds to make.

Show the answer and mark scheme
  • B1 cube

Answer: cube

Practice question (Core, non-calculator, 3 marks):

The net of a cuboid has a total area of \(190\text{ cm}^2\). The base of the cuboid is 5 cm by 7 cm. Find the height of the cuboid.

Show the answer and mark scheme
  • M1 2 × (35 + 5h + 7h) = 190
  • M1 35 + 12h = 95 or 24h = 120
  • A1 5 cm

Answer: \(5\) cm

Edexcel 4MA1 notes: Geometric Constructions
Practise the shared part (4MA1 questions): Bearings and Constructions questions

4.3 Scale drawings

Core C4.3 Extended E4.3

  • Draw and interpret scale drawings
  • Use and interpret three-figure bearings

Same mathematics as Edexcel 4MA1.

Revise it

From our Edexcel 4MA1 Higher notes, so it may go beyond Core: Core students need only the Core statements above.

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°.

0580 practice questions

6 questions written for 0580 in the Geometry practice: 6 Core, 3 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

The scale of a map is \(1 : 50\,000\). Two villages are 4 cm apart on the map. Work out the real distance between them in kilometres.

Show the answer and mark scheme
  • M1 4 × 50 000 = 200 000 cm
  • A1 2 km

Answer: \(2\) km

Practice question (Core, non-calculator, 2 marks):

The bearing of \(B\) from \(A\) is \(070^\circ\). Work out the bearing of \(A\) from \(B\).

Show the answer and mark scheme
  • M1 70 + 180
  • A1 250°

Answer: \(250^\circ\)

Edexcel 4MA1 notes: Scale Drawings & Bearings
Practise (4MA1 questions, same mathematics): Bearings and Constructions questions

4.4 Similarity

Core C4.4 Extended E4.4

  • Calculate lengths of similar shapes

Extended also: Areas of similar shapes and surface areas and volumes of similar solids; Solve problems and give simple explanations involving similarity.

Same mathematics as Edexcel 4MA1.

0580 practice questions

6 questions written for 0580 in the Geometry practice: 6 Core, 3 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

Two rectangles are similar. The smaller one is 4 cm wide and 6 cm long. The larger one is 10 cm wide. Find the length of the larger rectangle.

Show the answer and mark scheme
  • M1 scale factor 10 ÷ 4 = 2.5
  • A1 15 cm

Answer: \(15\) cm

Practice question (Core, non-calculator, 2 marks):

Triangles \(ABC\) and \(DEF\) are similar, with \(A\) corresponding to \(D\), \(B\) to \(E\) and \(C\) to \(F\). \(AB = 5\) cm, \(DE = 15\) cm and \(BC = 7\) cm. Find \(EF\).

Show the answer and mark scheme
  • M1 scale factor 3
  • A1 21 cm

Answer: \(21\) cm

Edexcel 4MA1 notes: Geometry and Trigonometry (all notes)
Practise (4MA1 questions, same mathematics): Similar Shapes questions

4.5 Symmetry

Core C4.5 Extended E4.5

  • Line symmetry and order of rotational symmetry of 2D shapes

Extended also: Planes of symmetry and axes of rotational symmetry of prisms, cylinders, pyramids and cones.

Partly in Edexcel 4MA1: 4MA1 covers 2D symmetry of quadrilaterals only; 3D symmetry is not taught.

Notes for 0580

In 2D, a line of symmetry reflects a shape onto itself, and the order of rotational symmetry is the number of times the shape fits onto itself in one full turn. A regular polygon with n sides has n lines of symmetry and rotational symmetry of order n.

In 3D (Extended), a plane of symmetry cuts a solid into two halves that are mirror images, and an axis of rotational symmetry is a line the solid can turn about to fit onto itself; its order is the number of fits in one full turn.

Example (Core, non-calculator): State the number of lines of symmetry and the order of rotational symmetry of a regular hexagon.

  1. A regular hexagon has 6 sides, so it has 6 lines of symmetry (3 through opposite vertices and 3 through the midpoints of opposite sides).
  2. It fits onto itself 6 times in a full turn.

Answer: 6 lines of symmetry; rotational symmetry of order 6

Example (Extended, non-calculator): A right pyramid has a square base. How many planes of symmetry does it have, and what is the order of rotational symmetry about its vertical axis?

  1. Each of the 4 lines of symmetry of the square base, together with the apex above its centre, gives a vertical plane of symmetry.
  2. Turning about the vertical axis, it fits onto itself 4 times.

Answer: 4 planes of symmetry; order 4

Example (Extended, non-calculator): How many planes of symmetry does a cuboid measuring 2 cm by 3 cm by 5 cm have?

  1. Its faces are rectangles that are not squares, so the only planes of symmetry are the 3 that cut each dimension in half (the planes through opposite diagonal edges are not symmetry planes).

Answer: 3

0580 practice questions

12 questions written for 0580 in the Geometry practice: 7 Core and 5 Extended, all non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

Write down the number of lines of symmetry and the order of rotational symmetry of

(a) a square (b) a parallelogram that is not a rectangle or a rhombus.

Show the answer and mark scheme
  • B1 (a) 4 lines, order 4
  • B1 (b) 0 lines, order 2

Answer: (a) 4 lines, order 4 (b) 0 lines, order 2

Practice question (Extended, non-calculator, 2 marks):

A prism has a cross-section that is an equilateral triangle.

(a) How many planes of symmetry does the prism have?

(b) What is the order of rotational symmetry about the axis through the centres of the two triangular faces?

Show the answer and mark scheme
  • B1 (a) 4
  • B1 (b) 3

Answer: (a) 4 (b) 3

Edexcel 4MA1 notes: Geometry of Shapes
Practise the shared part (4MA1 questions): Quadrilaterals and Symmetry questions

4.6 Angles

Core C4.6 Extended E4.6

  • Angles on a line, at a point, vertically opposite, in triangles and quadrilaterals
  • Angles with parallel lines (corresponding, alternate, co-interior)
  • Interior and exterior angles of polygons

Same mathematics as Edexcel 4MA1.

Revise it

From our Edexcel 4MA1 Higher notes, so it may go beyond Core: Core students need only the Core statements above.

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°.

0580 practice questions

6 questions written for 0580 in the Geometry practice: 6 Core, 3 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

(a) Two angles on a straight line are \(47^\circ\) and \(x^\circ\). Find \(x\).

(b) Three angles at a point are \(110^\circ\), \(95^\circ\) and \(y^\circ\). Find \(y\).

Show the answer and mark scheme
  • B1 (a) 133
  • B1 (b) 155

Answer: (a) \(133\) (b) \(155\)

Practice question (Core, non-calculator, 2 marks):

In an isosceles triangle, the angle between the two equal sides is \(40^\circ\). Find the other two angles.

Show the answer and mark scheme
  • M1 (180 − 40) ÷ 2
  • A1 70° and 70°

Answer: \(70^\circ\) and \(70^\circ\)

Edexcel 4MA1 notes: Geometry of Shapes · Angles in Polygons
Practise (4MA1 questions, same mathematics): Angles and Polygons questions

4.7 Circle theorems I

Core C4.7 Extended E4.7

Core calls this sub-topic "Circle theorems"; "Circle theorems I" is the Extended title.

  • The angle in a semicircle is 90°
  • The angle between a tangent and a radius is 90°

Extended also: The angle at the centre is twice the angle at the circumference; Angles in the same segment are equal; Opposite angles of a cyclic quadrilateral add to 180°; The alternate segment theorem.

Same mathematics as Edexcel 4MA1.

Revise it

From our Edexcel 4MA1 Higher notes, so it may go beyond Core: Core students need only the Core statements above.

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°.

0580 practice questions

6 questions written for 0580 in the Geometry practice: 6 Core, 3 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

Practice question (Core, non-calculator, 2 marks):

\(AB\) is a diameter of a circle and \(C\) is a point on the circle. Angle \(CAB = 35^\circ\). Find angle \(ABC\).

Show the answer and mark scheme
  • B1 angle ACB = 90° (angle in a semicircle)
  • B1 55°

Answer: \(55^\circ\)

Practice question (Core, non-calculator, 2 marks):

\(PT\) is a tangent to a circle with centre \(O\), touching it at \(T\).

(a) Write down the size of angle \(OTP\).

(b) Angle \(TOP = 58^\circ\). Find angle \(OPT\).

Show the answer and mark scheme
  • B1 (a) 90° (tangent ⟂ radius)
  • B1 (b) 32°

Answer: (a) \(90^\circ\) (b) \(32^\circ\)

Edexcel 4MA1 notes: Circle Theorems
Practise (4MA1 questions, same mathematics): Circle Theorems questions

4.8 Circle theorems II

Extended only E4.8

Extended: Equal chords are equidistant from the centre; The perpendicular bisector of a chord passes through the centre; Tangents from an external point are equal in length.

Same mathematics as Edexcel 4MA1.

Edexcel 4MA1 notes: Circle Theorems
Practise (4MA1 questions, same mathematics): Circle Theorems questions

Practise geometry

60 practice questions are written for 0580 on 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7. For the 5 sub-topics that are the same mathematics in Edexcel 4MA1, the practice also has our 4MA1 Higher questions, which are Extended level. Core students: use the Core filter on the practice page; every sub-topic with Core content has questions written for 0580 Core. Practice is part of the IGCSE plan, with a free preview; these notes are free.