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Pythagoras and trigonometry: OCR GCSE Maths questions

Every OCR GCSE Maths exam question on G20 Pythagoras and trigonometry, 2017–2025, with its mark scheme and examiner’s report.

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921 questions

  1. Q3

    June 2025 · Paper 4

    4 marksG17 Circles, surface area and volumeG20 Pythagoras and trigonometry
    The whole paper ↗
    June 2025 Paper 4, question 3: The diagram shows a shape formed from a rectangle measuring 8 cm by 17 cm with a semicircle removed from one side. 4 cm 3 cm Not to scale 8 cm 17 cm Calculate t

  2. Q12

    June 2025 · Paper 4

    5 marksG16 Area of 2D shapes and volume of prismsG20 Pythagoras and trigonometry
    The whole paper ↗
    June 2025 Paper 4, question 12: The diagram shows a trapezium. 17.2 cm Not to scale 7 cm 7 cm 25.6 cm Calculate the area of the trapezium. You must show your working. 2 cm

  3. Q22

    June 2025 · Paper 5

    6 marksG23 Area of a triangle (½ab sin C)G20 Pythagoras and trigonometry
    The whole paper ↗
    June 2025 Paper 5, question 22: The diagram shows a quadrilateral ABCD. DB is a diagonal of the quadrilateral. C Not to scale 14 cm D 5 cm 45° x° A B 7 cm AD = 5 cm, AB = 7 cm and BC = 14 cm.

  4. Q22

    June 2025 · Paper 6

    6 marksG20 Pythagoras and trigonometryG12 Properties of 3D shapes
    The whole paper ↗
    June 2025 Paper 6, question 22: The diagram shows a square-based pyramid ABCDE. O is the centre of the base. E D C O 10.7 cm 67° A B 10.7 cm The length of the base is 10.7 cm. Angle EAO is 67°

  5. Q21

    November 2024 · Paper 6

    5 marksG20 Pythagoras and trigonometry
    The whole paper ↗
    November 2024 Paper 6, question 21: The diagram shows a cuboid ABCDEFGH. A B D C Not to scale 15 cm E F 7 cm H G 24 cm FB = 15 cm, GF = 7 cm and HG = 24 cm. Calculate the angle BHF. You must show

  6. Q14

    June 2024 · Paper 4

    6 marksG17 Circles, surface area and volumeG20 Pythagoras and trigonometry
    The whole paper ↗
    June 2024 Paper 4, question 14: The diagram shows a circle, centre O and radius 52 cm. Not to scale A B 52 cm O 72 cm T AB is a chord of the circle. The line through T is a tangent to the circ

  7. Q20

    June 2024 · Paper 5

    7 marksG18 Arcs and sectorsG20 Pythagoras and trigonometry
    The whole paper ↗
    June 2024 Paper 5, question 20: The shape below is part of a circle, centre O and radius 12 cm. Angle OAB = 30°. B Not to scale O 12 cm 30° A Work out the perimeter of the shape. Give your ans

  8. Q5

    June 2024 · Paper 6

    9 marksG17 Circles, surface area and volumeG20 Pythagoras and trigonometry
    The whole paper ↗
    June 2024 Paper 6, question 5: The diagram shows a regular pentagon made using ten congruent right-angled triangles. The length of one side of the pentagon is 12 cm. Not to scale 12 cm 2, cor

  9. Q14

    June 2024 · Paper 6

    5 marksG11 Geometry on coordinate axesG20 Pythagoras and trigonometry
    The whole paper ↗
    June 2024 Paper 6, question 14: The sketch shows the coordinate axes and the point (0, 10). y Not to scale 10 x O The distance from the point (0, 10) to a point (12, t) is 12.5 units. Work out

  10. Q10

    November 2023 · Paper 4

    6 marksG16 Area of 2D shapes and volume of prismsG20 Pythagoras and trigonometry
    The whole paper ↗
    November 2023 Paper 4, question 10: The diagram shows a pentagon ABCDE. 48 cm A E Not to scale 42.5 cm D 54 cm 34 cm B C Find the area of the pentagon. You must show your working. 2 cm

  11. Q23

    November 2023 · Paper 5

    9 marksG18 Arcs and sectorsG20 Pythagoras and trigonometry
    The whole paper ↗
    November 2023 Paper 5, question 23: In the diagram, • ABC is a right-angled triangle • ACD is the sector of a circle with centre A. D C Not to scale 45° 30° A B 4 m 8 r m2. (a) Show that the area

  12. Q9

    June 2023 · Paper 4

    8 marksA21 Forming and solving equationsG20 Pythagoras and trigonometry
    The whole paper ↗
    June 2023 Paper 4, question 9: The sides of this triangle are given in centimetres. The perimeter of the triangle is 80 cm. Not to scale 3x + 1 x + 5 2x + 8 (a) Find the length of each side o

  13. Q13

    June 2023 · Paper 5

    6 marksG20 Pythagoras and trigonometryN8 Exact calculation and surds
    The whole paper ↗
    June 2023 Paper 5, question 13: The diagram shows a triangle, ABC, with perpendicular height BD. Not to scale B 12 cm 45° 30° A C D BC = 12 cm, angle BCD = 30° and angle BAD = 45°. (a) Work ou

  14. Q8

    June 2023 · Paper 6

    4 marksG20 Pythagoras and trigonometryG3 Angles and polygons
    The whole paper ↗
    June 2023 Paper 6, question 8: Taylor designs a logo using isosceles triangles joined at a central point, P. This is the start of Taylor’s design. Not to scale P The completed design will hav

  15. Q14

    June 2023 · Paper 6

    7 marksG17 Circles, surface area and volumeG20 Pythagoras and trigonometry
    The whole paper ↗
    June 2023 Paper 6, question 14: The diagram shows a square-based pyramid ABCDE. O is the centre of the base. E 14.5 cm 14.5 cm D C O 20 cm A 20 cm B The pyramid has base length 20 cm and each

  16. Q11

    November 2022 · Paper 4

    6 marksG20 Pythagoras and trigonometryN15 Rounding and error intervals
    The whole paper ↗
    November 2022 Paper 4, question 11: The diagram shows two right-angled triangles that are joined together. All measurements are given accurate to 2 significant figures. 8.3 cm Not to scale x cm 42

  17. Q20

    November 2022 · Paper 5

    4 marksG20 Pythagoras and trigonometry
    The whole paper ↗
    November 2022 Paper 5, question 20: Kai has a box in the shape of a cuboid. The internal dimensions of the box are 10 cm by 4 cm by 6 cm. 6 cm 4 cm 10 cm Kai is given a pencil of length 13 cm. Sho

  18. Q9

    June 2022 · Paper 4

    5 marksG20 Pythagoras and trigonometryG17 Circles, surface area and volume
    The whole paper ↗
    June 2022 Paper 4, question 9: Points A, B and C lie on the circumference of a circle, centre O. Not to scale A O 6.4 m B 3.7 m C Angle ACB = 90°, AC = 6.4 m and BC = 3.7 m. Work out the circ

  19. Q15

    June 2022 · Paper 5

    4 marksG21 Exact trigonometric valuesG20 Pythagoras and trigonometry
    The whole paper ↗
    June 2022 Paper 5, question 15: ABC is an equilateral triangle of side length 2 cm. M is the midpoint of AC. B Not to scale 2 cm 2 cm A M C 2 cm 1 . Using this diagram, show that tan 30° = 3

  20. Q21

    June 2022 · Paper 6

    6 marksG20 Pythagoras and trigonometry
    The whole paper ↗
    June 2022 Paper 6, question 21: The diagram shows triangle ABC. X lies on BC such that angle AXC = 90 ° . A Not to scale 32° 43° B C X 7.5 cm BC = 7 5 . cm, angle ABC = 32 ° and angle ACB = 43
All 41 questions on Pythagoras and trigonometry, as a list
  1. June 2025, Paper 4, Q3 (4 marks): The diagram shows a shape formed from a rectangle measuring 8 cm by 17 cm with a semicircle removed from one side. 4 cm 3 cm Not to scale 8 cm 17 cm Calculate t
  2. June 2025, Paper 4, Q12 (5 marks): The diagram shows a trapezium. 17.2 cm Not to scale 7 cm 7 cm 25.6 cm Calculate the area of the trapezium. You must show your working. 2 cm
  3. June 2025, Paper 5, Q22 (6 marks): The diagram shows a quadrilateral ABCD. DB is a diagonal of the quadrilateral. C Not to scale 14 cm D 5 cm 45° x° A B 7 cm AD = 5 cm, AB = 7 cm and BC = 14 cm.
  4. June 2025, Paper 6, Q22 (6 marks): The diagram shows a square-based pyramid ABCDE. O is the centre of the base. E D C O 10.7 cm 67° A B 10.7 cm The length of the base is 10.7 cm. Angle EAO is 67°
  5. November 2024, Paper 6, Q21 (5 marks): The diagram shows a cuboid ABCDEFGH. A B D C Not to scale 15 cm E F 7 cm H G 24 cm FB = 15 cm, GF = 7 cm and HG = 24 cm. Calculate the angle BHF. You must show
  6. June 2024, Paper 4, Q14 (6 marks): The diagram shows a circle, centre O and radius 52 cm. Not to scale A B 52 cm O 72 cm T AB is a chord of the circle. The line through T is a tangent to the circ
  7. June 2024, Paper 5, Q20 (7 marks): The shape below is part of a circle, centre O and radius 12 cm. Angle OAB = 30°. B Not to scale O 12 cm 30° A Work out the perimeter of the shape. Give your ans
  8. June 2024, Paper 6, Q5 (9 marks): The diagram shows a regular pentagon made using ten congruent right-angled triangles. The length of one side of the pentagon is 12 cm. Not to scale 12 cm 2, cor
  9. June 2024, Paper 6, Q14 (5 marks): The sketch shows the coordinate axes and the point (0, 10). y Not to scale 10 x O The distance from the point (0, 10) to a point (12, t) is 12.5 units. Work out
  10. November 2023, Paper 4, Q10 (6 marks): The diagram shows a pentagon ABCDE. 48 cm A E Not to scale 42.5 cm D 54 cm 34 cm B C Find the area of the pentagon. You must show your working. 2 cm
  11. November 2023, Paper 5, Q23 (9 marks): In the diagram, • ABC is a right-angled triangle • ACD is the sector of a circle with centre A. D C Not to scale 45° 30° A B 4 m 8 r m2. (a) Show that the area
  12. June 2023, Paper 4, Q9 (8 marks): The sides of this triangle are given in centimetres. The perimeter of the triangle is 80 cm. Not to scale 3x + 1 x + 5 2x + 8 (a) Find the length of each side o
  13. June 2023, Paper 5, Q13 (6 marks): The diagram shows a triangle, ABC, with perpendicular height BD. Not to scale B 12 cm 45° 30° A C D BC = 12 cm, angle BCD = 30° and angle BAD = 45°. (a) Work ou
  14. June 2023, Paper 6, Q8 (4 marks): Taylor designs a logo using isosceles triangles joined at a central point, P. This is the start of Taylor’s design. Not to scale P The completed design will hav
  15. June 2023, Paper 6, Q14 (7 marks): The diagram shows a square-based pyramid ABCDE. O is the centre of the base. E 14.5 cm 14.5 cm D C O 20 cm A 20 cm B The pyramid has base length 20 cm and each
  16. November 2022, Paper 4, Q11 (6 marks): The diagram shows two right-angled triangles that are joined together. All measurements are given accurate to 2 significant figures. 8.3 cm Not to scale x cm 42
  17. November 2022, Paper 5, Q20 (4 marks): Kai has a box in the shape of a cuboid. The internal dimensions of the box are 10 cm by 4 cm by 6 cm. 6 cm 4 cm 10 cm Kai is given a pencil of length 13 cm. Sho
  18. June 2022, Paper 4, Q9 (5 marks): Points A, B and C lie on the circumference of a circle, centre O. Not to scale A O 6.4 m B 3.7 m C Angle ACB = 90°, AC = 6.4 m and BC = 3.7 m. Work out the circ
  19. June 2022, Paper 5, Q15 (4 marks): ABC is an equilateral triangle of side length 2 cm. M is the midpoint of AC. B Not to scale 2 cm 2 cm A M C 2 cm 1 . Using this diagram, show that tan 30° = 3
  20. June 2022, Paper 6, Q21 (6 marks): The diagram shows triangle ABC. X lies on BC such that angle AXC = 90 ° . A Not to scale 32° 43° B C X 7.5 cm BC = 7 5 . cm, angle ABC = 32 ° and angle ACB = 43
  21. November 2021, Paper 4, Q19 (9 marks): ABC is an isosceles triangle. The sides of the triangle ABC are all tangents to a circle of radius 6 cm, centre O. B Not to scale 6 cm O 70° A C Angle BAC = 70°
  22. November 2021, Paper 5, Q11 (7 marks): The diagram shows a quadrilateral ABCD. AD = 10 cm, angle BAD = 30° and angle ABD = angle BDC = 90°. D C 10 cm Not to scale 30° A B The ratio of length BD to le
  23. November 2021, Paper 6, Q17 (5 marks): The diagram shows a pyramid ABCDE. E Not to scale 6.8 cm D C O 5.6 cm A B 5.6 cm The pyramid has a square horizontal base ABCD with side 5.6 cm. The vertex E is
  24. November 2020, Paper 5, Q20 (7 marks): ABCD is a trapezium. D C Not to scale A B The perimeter of the trapezium is 56 cm. The ratio AD : AB : DC : BC = 5 : 12 : 6 : 5. Calculate the area of the trape
  25. November 2019, Paper 4, Q18 (8 marks): The diagram shows a right-angled triangular prism ABCDEF. E F 6 cm B C 10 cm A D 11 cm Length AD = 11 cm, length CD = 10 cm and length CF = 6 cm. (a) Calculate
  26. November 2019, Paper 5, Q19 (7 marks): (a) Write down the value of sin 45°. (a) (b) ADB and BCD are right-angled triangles. BC = CD. AD = 10 6 mm. Angle BAD = 30°. 1 tan 30° = 3 B Not to scale C 30°
  27. June 2019, Paper 4, Q7 (6 marks): The diagram shows a shape ABCDE. The shape is made from a rectangle, a right-angled triangle and a quarter of a circle. C Not to scale B D F 41 m A E 18 m F is
  28. June 2019, Paper 5, Q15 (6 marks): OAB is a sector of a circle, centre O. OA = 6 cm and AX is perpendicular to OB. A Not to scale 6 cm O X B The area of sector OAB is 6r cm2. Show that AX = 3 3 c
  29. June 2019, Paper 6, Q5 (3 marks): ABC is a right-angled triangle. AB = 20 cm and BC = 37 cm. 37 cm C B Not to scale 20 cm A Calculate angle BAC. °
  30. June 2019, Paper 6, Q14 (5 marks): The length of the longest diagonal of a cube is 25 cm. Calculate the total surface area of the cube. cm2
  31. November 2018, Paper 4, Q3 (1 mark): Here is Mario’s answer to a question. Question 3 Answer 92 – 62 x = 9 mm x mm x = 45 x = 6.708( d.p.) 3 88° 43° 6 mm Work out the value of x. Explain the error
  32. November 2018, Paper 4, Q20 (6 marks): The diagram shows a right-angled triangular prism ABCDEF. D A E 18 cm F 40 cm B C 25 cm Calculate angle AFB. °
  33. November 2018, Paper 5, Q13 (8 marks): In the diagram, ABC is a right-angled triangle. P is a point on AB. BC = 40 m, AP = 20 m and angle ABC = 30 °. C 40 m Not to scale 30° A B P 20 m (a) Show that
  34. June 2018, Paper 4, Q8 (5 marks): A, B, C and D are points on the circumference of a circle, centre O. C Not to scale 6.4 cm B O D 28° A Angle CAD = 28° and CD = 6.4 cm. BD is a diameter of the
  35. June 2018, Paper 5, Q11 (6 marks): The diagram shows two right-angled triangles ABD and BCD, sharing a common side BD. AD = 10 cm, BC = 12 cm and angle DBC = 60°. B A Not to scale 60° 12 cm 10 cm
  36. November 2017, Paper 4, Q14 (5 marks): The diagram shows a cuboid ABCDEFGH. A B D C 8 cm E F 6 cm H G 11 cm Calculate angle BHF. °
  37. November 2017, Paper 4, Q17 (7 marks): ABC and ACD are triangles. Not to scale A 18 cm 78° D 46° 14 cm 81° C B (a) Show that AC = 13.0 cm, correct to 3 significant figures. (b) Calculate BC. (b) cm
  38. November 2017, Paper 5, Q8 (6 marks): The angles in a triangle are in the ratio 1 : 2 : 3. (a) Show that the triangle is a right-angled triangle. (b) The hypotenuse of the triangle is 15 cm long. Ca
  39. June 2017, Paper 4, Q7 (6 marks): A, B, C and D are four towns. North Not to scale B is 25 kilometres due East of A. C C is 25 kilometres due North of A. D is 45 kilometres due South of A. A B D
  40. June 2017, Paper 5, Q20 (7 marks): The diagram shows some land in the shape of a quadrilateral, ABCD. B Not to scale 3 km 30° A C 5 km 12 km D AB = 3 km, AD = 5 km, CD = 12 km and angle BAC = 30°
  41. June 2017, Paper 6, Q18 (7 marks): Alvin has a crate in the shape of a cuboid. The crate is open at the top. The internal dimensions of the crate are 46 cm long by 46 cm wide by 55 cm high. 55 cm

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