Geometry — IGCSE Mathematics (0580) questions by topic
Geometry questions here work out one interior angle of a 15-sided regular polygon, find which point on a folded net joins with a labelled point, use exterior angles marked on an equilateral triangle to find an unknown value, and reason from mathematically similar triangles with given side lengths. Find, calculate, draw, state and deduce want geometric reasoning shown step by step, since a correct final answer without justification often does not earn full marks.
93 questions from 2023 to 2025 make up this set, worth between 1 and 18 marks. Because similar-triangle and net-folding questions are marked on the reasoning chain, not just the answer, submitting your working for instant marking shows exactly where a ratio was misapplied or a fold was traced incorrectly, before that single step undermines an otherwise correct approach.
May–June 2025, Paper 11
The diagram shows a cuboid measuring 6cm by 4cm by 3cm (NOT TO SCALE). Complete a net of this cuboid on the 1cm^2 grid. One face has been drawn for you (a 6cm by 4cm rectangle).
Answer this question and get it marked →The diagram shows two straight lines intersecting two parallel lines. Find the value of x. Give a geometrical reason for your answer.
Answer this question and get it marked →Points A, B and C lie on the circle, centre O. The diagram shows the circle with AC as a diameter through the centre O, with B on the circle. Angle BAC (the angle at A between…
Answer this question and get it marked →May–June 2025, Paper 12
The diagram shows a line AB and a point P. The point P lies on the line AB, nearer to B than to A. (Diagram: a straight line sloping gently downwards from A on the upper left to B…
Answer this question and get it marked →A cuboid has length 5 cm, width 2 cm and height 3 cm. Draw a net of the cuboid on the 1 cm^2 grid.
Answer this question and get it marked →A quadrilateral has the geometrical properties - 4 equal length sides - 2 lines of symmetry - rotational symmetry of order 2. Write down the mathematical name of this…
Answer this question and get it marked →May–June 2025, Paper 13
Write down the mathematical name for an angle between 90° and 180°.
Answer this question and get it marked →The diagram shows a quadrilateral with sides of equal length. The bottom-left interior angle is a° and the bottom-right interior angle is 2a°. Write down the mathematical name for…
Answer this question and get it marked →The diagram shows a circle, centre O. P and S are points on the circle. POR is a straight line. QRST is a tangent to the circle at S. Angle OPS = 25°, angle x° is marked at S…
Answer this question and get it marked →Rectangle A has width 8 cm and height x cm. Rectangle B has width 2 cm and height 1.5 cm. Rectangle A is mathematically similar to rectangle B. Work out the value of x.
Answer this question and get it marked →May–June 2025, Paper 21
The diagram shows two parallel lines intersecting two straight lines. The two parallel lines (each marked with an arrow) cross two other straight lines that meet at a point. At…
Answer this question and get it marked →The interior angle of a regular polygon is 150°. Find the number of sides of this polygon.
Answer this question and get it marked →A, B and C are three points on a circle, centre O. DE is a tangent to the circle at A. Angle ACO = 35° and angle BCO = 40°. The diagram shows the circle with centre O; A is at the…
Answer this question and get it marked →May–June 2025, Paper 22
The diagram shows a 4 by 4 grid of small squares. In the top row, the 2nd and 4th squares (from the left) are shaded grey. In the bottom row, the 1st square (leftmost) is shaded…
Answer this question and get it marked →The scale drawing shows the positions of two villages, P and Q. The scale is 1 cm represents 0.5 km. The drawing shows P with a North arrow above it, and Q lower and to the right…
Answer this question and get it marked →The diagram (NOT TO SCALE) shows two straight lines intersecting two parallel lines. On the upper parallel line, the left intersection makes an angle of 45 degrees and the right…
Answer this question and get it marked →The diagram (NOT TO SCALE) shows a circle with points P, Q and R on the circumference. QR is a diameter passing through the circle. R is at the top left, P is at the right, and Q…
Answer this question and get it marked →May–June 2025, Paper 23
The diagram (not to scale) shows quadrilateral ABCD with various lines extended. ABCD is a quadrilateral. CDE is a straight line. AFB is an isosceles triangle. At vertex A, angle…
Answer this question and get it marked →The diagram shows a pyramid ABCDE with a square base. M is the centre of the square base. E is vertically above M. Write down the number of planes of symmetry of this pyramid.
Answer this question and get it marked →A, B, C and D are points on a circle. EF is a tangent to the circle at A. AB is parallel to DC. Find angle DCB, giving a geometrical reason.
Answer this question and get it marked →Solid A is mathematically similar to solid B. The diagram (NOT TO SCALE) shows two similar triangular prisms: Solid A is the smaller, with height 7 cm; Solid B is the larger, with…
Answer this question and get it marked →May–June 2025, Paper 31
An angle measures 157°. Write down the mathematical name for this type of angle.
Answer this question and get it marked →The diagram shows a jigsaw-puzzle-piece shape (a square outline with two rounded knobs/tabs protruding on adjacent outer edges and two matching indents on the other two edges).…
Answer this question and get it marked →In triangle ABC, AC = 6.4 cm and BC = 5.3 cm. Using a ruler and compasses only, construct triangle ABC. Leave in your construction arcs. The line AB has been drawn for you (a…
Answer this question and get it marked →The scale drawing shows the position of ship A. The scale is 1 centimetre represents 50 kilometres. A North arrow is drawn pointing upwards from point A. Ship B is 300 km from…
Answer this question and get it marked →Two triangles are shown (NOT TO SCALE). Triangle ABC has base AC = 12 cm with vertex B above. Triangle UVW has base UW = 21 cm with vertex V above, and side UV = 14 cm. Triangle…
Answer this question and get it marked →May–June 2025, Paper 32
A, B and C are points on a circle, centre O, diameter AC. Complete these statements, giving geometrical reasons. The value of x is .......................... because…
Answer this question and get it marked →The scale drawing shows the position of a hospital, H. The scale is 1 centimetre represents 6 kilometres. A North arrow points vertically up from H, with H marked at the foot of…
Answer this question and get it marked →May–June 2025, Paper 33
The diagram shows a grid of 6 columns by 4 rows of unit squares, with some squares shaded grey. The shaded squares form a pattern: in the top row the two end squares (columns 1…
Answer this question and get it marked →ABC is a straight line. BD is parallel to AE. Find the value of x. Give a geometrical reason for your answer.
Answer this question and get it marked →The scale drawing shows a path from S to P. The scale is 1 cm represents 2.5 km. Work out the actual distance between S and P.
Answer this question and get it marked →The diagram shows a circle with two points C and D on the circle joined by a straight line CD (a line drawn across the circle, not through the centre). Points C and D lie on the…
Answer this question and get it marked →May–June 2025, Paper 42
A quadrilateral has these properties: - the diagonals are the only lines of symmetry - it has rotational symmetry of order 2. Write down the mathematical name of this…
Answer this question and get it marked →The diagram shows the net of a solid. The net consists of a central rectangle (the base) with a square flap above it and another rectangle below it, and two triangular flaps…
Answer this question and get it marked →The diagram shows two mathematically similar triangles (not to scale). Triangle ABC has AB = 9 cm, AC = 12 cm and BC = 16 cm, with equal base angles marked at B and C. Triangle…
Answer this question and get it marked →The interior angle of a regular polygon is 172°. Find the number of sides of this polygon.
Answer this question and get it marked →The diagram shows five points A, B, C, D and E lying on a circle (not to scale), with chords drawn between them. At vertex C, angle DCE = 26° and angle ECB = 52°. At vertex E,…
Answer this question and get it marked →May–June 2025, Paper 43
In triangle ABC, AC = 21 m and BC = 15.9 m. Using a ruler and compasses only, complete the scale drawing of the triangle. Use a scale of 1 cm to represent 3 m. The side AB is…
Answer this question and get it marked →May–June 2023, Paper 11
The diagram shows a pair of parallel lines and a straight line, with one marked angle 52° and an angle x° in the alternate position. Write down the geometrical reason why the…
Answer this question and get it marked →The diagram shows a right-angled triangle with angles 4x° and (3x − 1)° (and the right angle). Use the information in the diagram to write down and solve an equation to find the…
Answer this question and get it marked →Triangle ABC is similar to triangle DEF. In ABC, AB = 5.6 cm and BC = 7.2 cm. In DEF, DE = h cm and EF = 8.1 cm. Calculate the value of h.
Answer this question and get it marked →May–June 2023, Paper 12
The diagram shows a line AB. Draw a line perpendicular to the line AB.
Answer this question and get it marked →The diagram shows a grid of squares, some shaded. Shade two squares so that the diagram has rotational symmetry of order 4.
Answer this question and get it marked →In the diagram, ABC is a triangle and ACD is a straight line. Angle ABC = 59°, angle BAC = 73°, angle BCA = x° and angle BCD = y°. Find the value of x and the value of y.
Answer this question and get it marked →The scale drawing shows the positions of town A and town B, with a North line at A. Measure the bearing of town B from town A.
Answer this question and get it marked →Find the size of an interior angle of a regular 15-sided polygon.
Answer this question and get it marked →Triangle ABC is similar to triangle PQR. AB = 15 cm, BC (=CB segment, the side 9 cm is AC) — specifically AC = 9 cm and AB = 15 cm; PQ = 18 cm corresponds to AB. Calculate QR.
Answer this question and get it marked →May–June 2023, Paper 13
The diagram shows a shape. Complete the statement: The diagram has rotational symmetry of order ...
Answer this question and get it marked →Triangle ABC is isosceles. Angle BAC = 38° and AB = AC. Angle ABC = x°. Find the value of x.
Answer this question and get it marked →The diagram shows a circle, centre O, diameter AB. A, B and C lie on the circumference. Write down the mathematical name of the line AC.
Answer this question and get it marked →May–June 2023, Paper 21
The diagram shows angles at points B, C and D with CDE a straight line. Angle ABC-region marks: angle at B = 104°, angle at A = 71°, angle at C = 56°. Find angle ADE.
Answer this question and get it marked →The diagram shows a straight line intersecting two parallel lines, with angle 58° marked, and angles a° and b°. Find the value of a and the value of b, giving a geometrical reason…
Answer this question and get it marked →Triangle ABC is similar to triangle DEF. In ABC, AB = 5.6 cm and BC = 7.2 cm; in DEF, DE = h cm and EF = 8.1 cm. Calculate the value of h.
Answer this question and get it marked →AO, OB and OC are all radii of the circle. AB = BC. Therefore triangle AOB is congruent to triangle COB. Draw a ring around the correct criterion for this statement (SAS, RHS,…
Answer this question and get it marked →May–June 2023, Paper 22
A, B, C and D are points on a circle. AB is parallel to DC and angle ACD = 32°. Chords AC and DB intersect at E, with angle AEB (or the marked angle) = x°. Find the value of x.
Answer this question and get it marked →May–June 2023, Paper 23
The diagram shows a shape. Complete the statement: The diagram has rotational symmetry of order ...
Answer this question and get it marked →May–June 2023, Paper 31
Complete this statement: The mathematical name of any polygon with 4 sides is a ...
Answer this question and get it marked →The diagram shows the net of a solid on a 1 cm² grid with labelled points A..N. When the net is folded to make the solid, point C joins with point A. Write down which other point…
Answer this question and get it marked →The diagram shows a circle, centre O, with B, D and E on the circumference. AOEF is a straight line and AC touches the circle at B. Write down the mathematical name for line BOD.
Answer this question and get it marked →May–June 2023, Paper 32
For this quadrilateral, draw any lines of symmetry and write down its mathematical name (the shape has two diagonals as lines of symmetry).
Answer this question and get it marked →May–June 2023, Paper 33
The diagram shows a triangle ABC and a straight line BCD. Angle BAC = 46°, angle ACB = 108°, angle ACD-region marks x° and y°. Angle ACB = 108°. Write down the mathematical name…
Answer this question and get it marked →May–June 2023, Paper 42
The diagram shows an isosceles triangle with the base extended. The marked angle at the apex is 42°, and x° is the exterior angle formed where the base is extended. NOT TO SCALE.…
Answer this question and get it marked →May–June 2023, Paper 43
The scale drawing shows two sides, AB and BC, of a field. The scale is 5 centimetres represents 200 metres. Measure angle ABC.
Answer this question and get it marked →A, B and C are three towns and the bearing of C from A is 114°. B is due south of A and AC = BC. NOT TO SCALE. Calculate the bearing of B from C.
Answer this question and get it marked →October–November 2023, Paper 11
The diagram shows an angle that is greater than 90° but less than 180°. Write down the mathematical name for this type of angle.
Answer this question and get it marked →The diagram shows an isosceles triangle. The apex angle is 41° and x° is one of the two equal base angles. NOT TO SCALE. Find the value of x.
Answer this question and get it marked →AB and CD are parallel lines. A transversal arrangement gives an angle of 52° at A (angle between AB and the line) with x° adjacent, and an angle of 135° at D on line CD. NOT TO…
Answer this question and get it marked →October–November 2023, Paper 12
The diagram shows two straight lines crossing two parallel lines. Angles of 50° and x° are marked at one intersection and 114° at another. NOT TO SCALE. Find the value of x.
Answer this question and get it marked →P and Q are two points. North lines are shown at both P and Q. The bearing of P from Q is given by an angle of 39° between the east direction at Q and the line QP. NOT TO SCALE.…
Answer this question and get it marked →Calculate the interior angle of a regular 9-sided polygon.
Answer this question and get it marked →A, B and C are points on a circle, centre O. Draw a tangent to the circle at point A.
Answer this question and get it marked →Triangle ABC is mathematically similar to triangle DEF. In triangle ABC, AB = 7.3 cm and AC = 5.5 cm. In triangle DEF, DF = 4.4 cm and DE = x cm (DE corresponds to AB). NOT TO…
Answer this question and get it marked →October–November 2023, Paper 13
The diagram shows a shaded shape and several other shapes labelled A, B, C, D and E. Write down the letter of the shape that is congruent to the shaded shape.
Answer this question and get it marked →Draw the lines of symmetry on this rhombus.
Answer this question and get it marked →In triangle LMN, LN = 7.5 cm and MN = 8 cm. Using a ruler and compasses only, construct triangle LMN. Leave in your construction arcs. The line LM has been drawn for you.
Answer this question and get it marked →The diagram shows an equilateral triangle. Two exterior angles, each x°, are marked. NOT TO SCALE. Find the value of x.
Answer this question and get it marked →October–November 2023, Paper 21
The diagram shows an isosceles triangle. The apex angle is 41° and x° is one of the two equal base angles. NOT TO SCALE. Find the value of x.
Answer this question and get it marked →A, B and C are points on the circumference of a circle, centre O. Tangent DE touches the circle at C. Angle BCE = 53° and angle ACO = 20°. NOT TO SCALE. Find the value of x, where…
Answer this question and get it marked →The diagram shows a cuboid ABCDEFGH. AB = 15.1 cm, BC = 4.5 cm and CG = 9.2 cm. NOT TO SCALE. Calculate the angle that the diagonal BH makes with the face ADHE.
Answer this question and get it marked →October–November 2023, Paper 22
The diagram shows two intersecting straight lines crossing two parallel lines. Angles of 50° and x° are marked at one intersection and 114° at another. NOT TO SCALE. Find the…
Answer this question and get it marked →P and Q are two points. North lines are shown at both P and Q. The bearing of P from Q is given by an angle of 39° between the east direction at Q and the line QP. NOT TO SCALE.…
Answer this question and get it marked →The two shapes are mathematically similar. The smaller shape has a side of 6.3 cm corresponding to 9 cm on the larger shape, and a side of 5 cm corresponding to h cm on the larger…
Answer this question and get it marked →A, B and C are points on a circle, centre O. Angle OBA = 28°. NOT TO SCALE. Find angle ACB.
Answer this question and get it marked →October–November 2023, Paper 31
The diagram shows a circle. The diameter of this circle is 168 mm. Write down the radius of this circle.
Answer this question and get it marked →The diagram shows a cuboid 5 cm by 3 cm by 2 cm. NOT TO SCALE. On the 1 cm² grid, complete the net of the cuboid. One face has been drawn for you.
Answer this question and get it marked →Triangle PQR is mathematically similar to triangle UVW. In PQR: PR = 21.4 cm, PQ = 9.6 cm, QR = 12.8 cm. In UVW: UW = 14.4 cm, with VW corresponding to QR. NOT TO SCALE. Calculate…
Answer this question and get it marked →October–November 2023, Paper 32
Write down the mathematical name for this polygon (a five-sided shape shown).
Answer this question and get it marked →In triangle RST, RT = 7 cm and ST = 4 cm. Using a ruler and compasses only, construct triangle RST. Leave in your construction arcs. The line RS has been drawn for you.
Answer this question and get it marked →October–November 2023, Paper 33
The scale drawing shows the positions of Bayside (B) and Millwater (M). The scale is 1 centimetre represents 2 kilometres. Find the actual distance between Bayside and Millwater.
Answer this question and get it marked →In the diagram, PST is a straight line. Angle QPS = 114°, angle PSR = 114°, angle QRS = y° and angle PQR = 3y°. NOT TO SCALE. Give the geometrical reason why the lines PQ and SR…
Answer this question and get it marked →The sketch shows the graph of y = x² − 2x − 3. The graph crosses the x-axis at (−1, 0) and (3, 0). NOT TO SCALE. Find the equation of the line of symmetry of the graph.
Answer this question and get it marked →October–November 2023, Paper 42
ABCDEF is a regular hexagon. DF, DA and DB are diagonals. NOT TO SCALE. Complete the following statements using three different triangles: Triangle DEF is congruent to triangle…
Answer this question and get it marked →October–November 2023, Paper 43
Find the size of one interior angle of a regular 10-sided polygon.
Answer this question and get it marked →Other IGCSE Mathematics topics
- Number (230)
- Algebra and graphs (150)
- Mensuration (54)
- Statistics (52)
- Trigonometry (33)
- Transformations and vectors (32)
- Probability (27)
- Coordinate geometry (16)