IGCSE Mathematics: Geometry — Practice Questions

These are original questions written for Marlbridge, in the style and at the standard of the examination. They are not reproduced past-paper questions — examination boards hold copyright in their own papers. Use these alongside the official past papers available free from your board.

Related: Geometry revision notes

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Questions

1. State the mathematical term for: (a) a line that divides an angle into two equal parts; (b) two shapes that have the same shape and size. [2]

2. Describe, step by step, how to construct the perpendicular bisector of a line segment AB using only a pair of compasses and a straight edge. [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) 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. 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]

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Answers

1. (a) Bisector (angle bisector) [1]. (b) Congruent [1].

2. Open the compasses to more than half the length of AB [1]. With the point on A, draw arcs above and below the line; repeat with the point on B, using the same radius, so the arcs intersect above and below AB [1]. Draw a straight line through the two intersection points — this is the perpendicular bisector [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].

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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.