IGCSE Mathematics: Geometry — Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs — examination boards hold copyright in their own papers. Use these alongside the official past papers available free from your board.

Related: Geometry revision notes

Tier note: items marked (Extended) are Extended only in the 0580 syllabus: area scale factors of similar shapes (E4.4; Core calculates lengths only) and the circle theorems beyond the Core pair, such as the angle at the centre and cyclic quadrilaterals (E4.7). The Core circle theorems are the angle in a semicircle and the angle between a tangent and a radius (C4.7). All other questions and parts use Core content.

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Questions

1. State the mathematical term for: (a) an angle that is greater than 180° but less than 360°; (b) two shapes that have the same shape and size. [2]

2. Describe, step by step, how to construct triangle ABC with AB = 7 cm, AC = 6 cm and BC = 5 cm, using only a ruler and a pair of compasses. [3]

3. A map is drawn to a scale of 1 : 25 000. The distance between two towns on the map is 4.4 cm. Calculate the real distance between the towns, in kilometres. [3]

4. Triangle ABC is similar to triangle PQR, with AB corresponding to PQ. AB = 2 cm and PQ = 5 cm. The area of triangle ABC is 12 cm².

(a) Find the linear scale factor from ABC to PQR. [1] (b) (Extended) Find the area of triangle PQR. [2]

5. (a) State the order of rotational symmetry of a regular hexagon. [1] (b) State the number of lines of symmetry of an isosceles triangle (that is not equilateral). [1]

6. A regular octagon has 8 equal sides and 8 equal angles.

(a) Calculate the size of one exterior angle. [2] (b) Calculate the size of one interior angle. [1]

7. (Extended) Points A, B and C lie on the circumference of a circle, centre O. The angle at the circumference, angle BAC, is 35°.

(a) State the circle theorem that relates angle BOC (the angle at the centre) to angle BAC. [1] (b) Calculate angle BOC. [1]

8. (Extended) PQRS is a cyclic quadrilateral. Angle P = 108°.

(a) State the circle theorem that lets you find angle R. [1] (b) Calculate angle R. [1] (c) A tangent touches the circle at point Q. Explain what is true about the angle between this tangent and the radius drawn to Q. [1]

9. Points D, E and F lie on the circumference of a circle. Angle DEF = 90°. State the circle theorem this confirms about the line DF, and explain your reasoning. [2]

10. Each interior angle of a regular polygon is 150°.

(a) Calculate the number of sides of the polygon. [2] (b) Calculate the sum of the interior angles of the polygon. [1]

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Answers

1. (a) Reflex angle [1]. (b) Congruent [1].

2. Draw AB = 7 cm with a ruler [1]. Open the compasses to 6 cm and, with the point on A, draw an arc; open them to 5 cm and, with the point on B, draw an arc that crosses the first arc [1]. The crossing point is C: join it to A and to B with a ruler, leaving both construction arcs visible [1].

3. Real distance = 4.4 × 25 000 = 110 000 cm [1] = 1100 m [1] = 1.1 km [1].

4. (a) Scale factor = 5 ÷ 2 = 2.5 [1]. (b) Area scale factor = 2.5² = 6.25 [1]. Area of PQR = 12 × 6.25 = 75 cm² [1].

5. (a) 6 [1]. (b) 1 [1].

6. (a) 360° ÷ 8 = 45° [2]. (b) 180° − 45° = 135° [1].

7. (a) The angle at the centre is twice the angle at the circumference, standing on the same arc [1]. (b) 2 × 35° = 70° [1].

8. (a) Opposite angles of a cyclic quadrilateral sum to 180° [1]. (b) 180° − 108° = 72° [1]. (c) The tangent and the radius meet at 90° (the tangent is perpendicular to the radius at the point of contact) [1].

9. This confirms the angle in a semicircle is 90° theorem [1]. Since angle DEF, standing on DF, is 90°, DF must be a diameter of the circle [1].

10. (a) Exterior angle = 180° − 150° = 30° [1]; number of sides = 360° ÷ 30° = 12 [1]. (b) Sum of interior angles = 12 × 150° = 1800° [1]. (Check: (12 − 2) × 180° = 1800°.)

Common mistake: dividing 360° by the interior angle (360° ÷ 150° = 2.4, which is not a whole number of sides). The exterior angles of any polygon add up to 360°, so divide 360° by the exterior angle.

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Where marks are usually lost

• Leaving construction arcs unmarked or not showing them at all — the method (visible arcs) is what earns the mark, not just a correctly-placed final line.
• Converting the map scale ratio in the wrong direction, or forgetting to convert cm to km at the end.
• Using the linear scale factor instead of its square when scaling an area (or its cube when scaling a volume).
• Confusing rotational symmetry (how many times a shape maps onto itself in one full turn) with reflective symmetry (lines of symmetry) — they are asked for separately and are not always equal.
• Quoting a circle theorem’s name only without applying the numbers, or vice versa — both are usually credited separately.
• Applying the cyclic-quadrilateral rule (“opposite angles sum to 180°”) to a general quadrilateral — it only holds for a cyclic quadrilateral, one whose four vertices all lie on a single circle; check that condition before using the rule.

Examiner report insight

• Applying the general quadrilateral angle sum (360 degrees) to a cyclic quadrilateral instead of the specific rule that opposite angles sum to 180 degrees – check whether all four vertices lie on a circle before choosing which rule applies.
• Using the alternate segment theorem without first justifying every angle relied on – a common error assumes a triangle in the figure is isosceles, or that two lines are parallel, without that being given.
• Assuming a kite has two lines of symmetry like a rhombus – a (non-rhombus) kite has exactly one line of symmetry, along the diagonal joining the two vertices where each pair of equal adjacent sides meets. That diagonal bisects the (generally unequal) angles at those two vertices; the kite’s one pair of equal angles sits at the other two vertices, off the axis, each the mirror image of the other rather than bisected by it.

Source: Cambridge International, 0580 Mathematics Principal Examiner Report, June 2024 series, Papers 13, 21, 23 (verified 2026-09-02).

Approaching geometry questions

For circle-theorem questions, the first task is always identifying which theorem applies – start by checking which points genuinely lie on the circle, whether a line passes through the centre, and whether a tangent is involved, since these features determine the theorem rather than the numbers themselves. For construction and scale-drawing questions, showing the method (visible arcs, a clearly labelled scale calculation) is what earns credit, not just a correct final answer or diagram, so treat the working as part of the answer rather than scratch work to be tidied away.