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

Cambridge IGCSE Maths 0580 Geometry Notes

Q: What is in Cambridge IGCSE 0580 geometry?

Topic 4 has 8 sub-topics: 4.1 Geometrical terms; 4.2 Geometrical constructions; 4.3 Scale drawings; 4.4 Similarity; 4.5 Symmetry; 4.6 Angles; 4.7 Circle theorems I; 4.8 Circle theorems II.

Q: Which geometry sub-topics are Extended only?

4.8 Circle theorems II. Core students (Papers 1 and 3) do not need them.

Q: What do students get wrong in geometry?

Map scales: not converting centimetres to kilometres; Bearings: the wrong angle, or measured from the wrong line. Our 0580 Examiner Insights page explains each one with a fix.

Q: Are these notes free?

Yes. Every note and worked example on these pages is free to read, with no sign-in.

Revision notes for topic 4, Geometry, of the Cambridge IGCSE Mathematics (0580) syllabus for exams from 2025 to 2030 (Cambridge says the 2028–2030 syllabus has no significant changes that affect teaching). 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.

8 sub-topics
7 Core and Extended
1 Extended only
3 with notes written for 0580

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?

(a) 2 triangles + 3 rectangles = 5 faces; 3 edges on each triangle + 3 joining them = 9 edges; 6 vertices. Check: 5 + 6 − 9 = 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

15 questions written for 0580 in the Geometry practice: 14 Core and 1 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 mathematical name for

(a) a straight line that touches a circle at exactly one point

(b) a straight line from the centre of a circle to its circumference.

Show the answer and mark scheme

B1 (a) tangent
B1 (b) radius

Answer: (a) tangent (b) radius

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

0580 questions: Geometrical Terms questions (Core and Extended, with worked answers)
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.

The net has two of each rectangle: 3 × 4, 3 × 5 and 4 × 5.
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.

The two shorter sides add to 4 + 5 = 9 cm, which is less than 10 cm.
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

15 questions written for 0580 in the Geometry practice: 15 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 one regular hexagon and six identical triangles. Write down the name of the solid it folds to make.

Show the answer and mark scheme

B1 hexagonal(-based) pyramid

Answer: hexagonal pyramid

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

A tin of soup is a closed cylinder with radius 4 cm and height 9 cm. Its net is made of two circles and one rectangle. Find the total area of the net, leaving $π$ in your answer.

Show the answer and mark scheme

M1 2 × π × 4² = 32π
M1 2π × 4 × 9 = 72π
A1 104π cm²

Answer: $104 π {cm}^{2}$

0580 questions: Geometrical Constructions questions (Core and Extended, with worked answers)
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

8 questions written for 0580 in the Geometry practice: 8 Core, 5 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, 3 marks):

The bearing of $B$ from $A$ is 128°.

(a) Work out the bearing of $A$ from $B$ .

(b) The bearing of $C$ from $A$ is 215°. Work out the size of angle $B A C$ .

Show the answer and mark scheme

M1 (a) 128 + 180
A1 308°
B1 (b) 215 − 128 = 87°

Answer: (a) $308^{\circ}$ (b) $87^{\circ}$

0580 questions: Scale Drawings and Bearings questions (Core and Extended, with worked answers)
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

8 questions written for 0580 in the Geometry practice: 7 Core and 1 Extended, 4 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 (Extended, calculator, 4 marks):

Two mathematically similar jugs have heights 15 cm and 24 cm.

(a) The smaller jug holds 0.5 litres. Work out how much the larger jug holds.

(b) The surface area of the larger jug is 640 cm². Work out the surface area of the smaller jug.

Show the answer and mark scheme

M1 (a) length scale factor 24/15 = 1.6 and volume factor 1.6³
A1 0.5 × 4.096 = 2.048 litres
M1 (b) 640 ÷ 1.6²
A1 250 cm²

Answer: (a) $2.048$ litres (b) $250$ cm²

0580 questions: Similar Shapes questions (Core and Extended, with worked answers)
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.

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).
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?

Each of the 4 lines of symmetry of the square base, together with the apex above its centre, gives a vertical plane of symmetry.
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?

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

15 questions written for 0580 in the Geometry practice: 9 Core and 6 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 rectangle that is not a square

(b) a kite that is not a rhombus.

Show the answer and mark scheme

B1 (a) 2 lines, order 2
B1 (b) 1 line, order 1

Answer: (a) $2$ lines, order $2$ (b) $1$ line, order $1$

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

An unsharpened pencil is a prism whose cross-section is a regular hexagon.

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

(b) What is the order of rotational symmetry about the axis along the middle of the pencil?

Show the answer and mark scheme

B1 (a) 7 (6 containing the long axis, 1 across the middle)
B1 (b) 6

Answer: (a) $7$ planes (b) order $6$

0580 questions: Symmetry questions (Core and Extended, with worked answers)
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

8 questions written for 0580 in the Geometry practice: 8 Core, 5 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, 4 marks):

Each interior angle of a regular polygon is 156°.

(a) How many sides does the polygon have?

(b) Work out the sum of its interior angles.

Show the answer and mark scheme

M1 (a) exterior angle 180 − 156 = 24°
A1 360 ÷ 24 = 15 sides
M1 (b) (15 − 2) × 180 or 15 × 156
A1 2340°

Answer: (a) $15$ (b) $2340^{\circ}$

0580 questions: Angles questions (Core and Extended, with worked answers)
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

8 questions written for 0580 in the Geometry practice: 7 Core and 1 Extended, 5 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):

$A B$ is a diameter of a circle and $C$ is a point on the circle. Angle $C A B = 35^{\circ}$ . Find angle $A B C$ .

Show the answer and mark scheme

B1 angle ACB = 90° (angle in a semicircle)
B1 55°

Answer: $55^{\circ}$

Practice question (Extended, non-calculator, 4 marks):

$A$ , $B$ , $C$ and $D$ lie on a circle with centre $O$ . Angle $A O C = 136^{\circ}$ . $B$ is on the major arc $A C$ and $D$ is on the minor arc $A C$ .

Work out (a) angle $A B C$ (b) angle $A D C$ . Give reasons.

Show the answer and mark scheme

M1 (a) angle at the centre is twice the angle at the circumference
A1 136 ÷ 2 = 68°
M1 (b) opposite angles of a cyclic quadrilateral add to 180°
A1 180 − 68 = 112°

Answer: (a) $68^{\circ}$ (b) $112^{\circ}$

0580 questions: Circle Theorems questions (Core and Extended, with worked answers)
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.

0580 practice questions

15 questions written for 0580 in the Geometry practice: 15 Extended, 8 non-calculator, each with a Cambridge-style mark scheme (M method, A accuracy, B independent marks). Two of them:

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

Two chords of a circle each have length 10 cm. One chord is 4 cm from the centre of the circle.

How far is the other chord from the centre? Give a reason.

Show the answer and mark scheme

B1 4 cm
B1 equal chords are equidistant from the centre

Answer: $4$ cm: equal chords are equidistant from the centre

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

$P A$ and $P B$ are tangents from the point $P$ to a circle with centre $O$ . Angle $A O B = 130^{\circ}$ .

Work out (a) angle $A P B$ (b) angle $P A B$ .

Show the answer and mark scheme

M1 (a) angles OAP and OBP are 90°: 360 − 90 − 90 − 130
A1 (a) 50°
B1 (b) PA = PB, so (180 − 50)/2 = 65°

Answer: (a) $50^{\circ}$ (b) $65^{\circ}$

0580 questions: Circle Theorems questions (Core and Extended, with worked answers)
Edexcel 4MA1 notes: Circle Theorems
Practise (4MA1 questions, same mathematics): Circle Theorems questions

Practise geometry

92 practice questions are written for 0580 on every sub-topic of this topic. 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.

Geometry practice for 0580: the 92 questions written for 0580, tagged Core or Extended and non-calculator or calculator, with the Edexcel 4MA1 Higher questions for the sub-topics that are the same mathematics
Student hub: the Edexcel 4MA1 units (Unit 5), each question marked
Cambridge IGCSE Maths 0580 past papers: the 2025 specimen papers and past papers, with a timer
Formulas for geometry on the one-page 0580 formula sheet (free PDF), showing which are on the paper's formula list
Cambridge 0580 geometry questions by topic, Core and Extended, with worked answers

More on this topic

Knowledge organiser: Geometry on one page (printable)

Knowledge organiser: Geometry on one page (key facts, formulas and mistakes to avoid; printable PDF)

Questions about geometry

What is in Cambridge IGCSE 0580 geometry?

Topic 4 has 8 sub-topics: 4.1 Geometrical terms; 4.2 Geometrical constructions; 4.3 Scale drawings; 4.4 Similarity; 4.5 Symmetry; 4.6 Angles; 4.7 Circle theorems I; 4.8 Circle theorems II.

Which geometry sub-topics are Extended only?

4.8 Circle theorems II. Core students (Papers 1 and 3) do not need them.

What do students get wrong in geometry?

Map scales: not converting centimetres to kilometres; Bearings: the wrong angle, or measured from the wrong line. Our 0580 Examiner Insights page explains each one with a fix.

Are these notes free?

Yes. Every note and worked example on these pages is free to read, with no sign-in.

Join free: practise geometry with marking →

Cambridge IGCSE Maths 0580 Geometry Notes

4.1 Geometrical terms

Notes for 0580

0580 practice questions

4.2 Geometrical constructions

Notes for 0580

0580 practice questions

4.3 Scale drawings

Revise it

0580 practice questions

4.4 Similarity

0580 practice questions

4.5 Symmetry

Notes for 0580

0580 practice questions

4.6 Angles

Revise it

0580 practice questions

4.7 Circle theorems I

Revise it

0580 practice questions

4.8 Circle theorems II

0580 practice questions

Practise geometry

Try it now

Questions about geometry