Mensuration and calculation

50 maths questions with worked answers with answers for G16–G23 Mensuration and calculation in Edexcel GCSE Maths (spec 1MA1). Practise with flashcards, a topic quiz, a random quiz across the course, 5 crosswords, or read every question and answer below. Not official Pearson material.

1. Question 1 · Area · G16

   Find the area of a triangle with base 12 cm and height 7 cm.

   Answer:

   Area = 12 × base × height.
   0.5 × 12 × 7 = 42 cm².

2. Question 2 · Area · G16

   Find the area of a parallelogram with base 9 cm and perpendicular height 5 cm.

   Answer:

   Area = base × height.
   45 cm².

3. Question 3 · Area · G16

   Find the area of a trapezium with parallel sides 6 cm and 10 cm and height 4 cm.

   Answer:

   Area = 12(a + b)h.
   0.5 × 16 × 4 = 32 cm².

4. Question 4 · Volume · G16

   Find the volume of a cuboid 5 cm by 4 cm by 3 cm.

   Answer:

   Volume = length × width × height.
   60 cm³.

5. Question 5 · Volume · G16

   Find the volume of a prism with cross-sectional area 18 cm² and length 10 cm.

   Answer:

   Volume = area of cross-section × length.
   180 cm³.

6. Question 6 · Surface area · G16

   Find the surface area of a cube with side 4 cm.

   Answer:

   6 faces each 16 cm².
   96 cm².

7. Question 7 · Surface area · G16

   Find the surface area of a cuboid 6 cm by 3 cm by 2 cm.

   Answer:

   2(6 × 3) + 2(6 × 2) + 2(3 × 2) = 36 + 24 + 12.
   72 cm².

8. Question 8 · Circles · G17

   Find the circumference of a circle with radius 5 cm, to 1 d.p.

   Answer:

   C = 2πr = 10π.
   31.4 cm.

9. Question 9 · Circles · G17

   Find the area of a circle with diameter 8 cm, to 1 d.p.

   Answer:

   r = 4, so A = π × 16.
   50.3 cm².

10. Question 10 · Circles · G17

    A circle has circumference 30 cm. Find its diameter to 1 d.p.

    Answer:

    d = C ÷ π.
    30 ÷ π = 9.5 cm.

11. Question 11 · Circles · G17

    Find the area of a semicircle with radius 6 cm in terms of π.

    Answer:

    Full circle = 36π.
    18π cm².

12. Question 12 · Cylinders · G17

    Find the volume of a cylinder with radius 3 cm and height 10 cm in terms of π.

    Answer:

    V = πr²h.
    π × 9 × 10 = 90π cm³.

13. Question 13 · Cones · G17 · Higher only

    Find the volume of a cone with radius 6 cm and height 5 cm in terms of π.

    Answer:

    V = 13πr²h.
    13 × π × 36 × 5 = 60π cm³.

14. Question 14 · Spheres · G17 · Higher only

    Find the volume of a sphere with radius 3 cm in terms of π.

    Answer:

    V = 43πr³.
    43 × π × 27 = 36π cm³.

15. Question 15 · Spheres · G17 · Higher only

    Find the surface area of a sphere with radius 5 cm in terms of π.

    Answer:

    SA = 4πr².
    4 × π × 25 = 100π cm².

16. Question 16 · Cones · G17 · Higher only

    Find the curved surface area of a cone with radius 4 cm and slant height 9 cm in terms of π.

    Answer:

    Curved SA = πrl.
    π × 4 × 9 = 36π cm².

17. Question 17 · Pyramids · G17 · Higher only

    Find the volume of a square-based pyramid with base 6 cm and height 10 cm.

    Answer:

    V = 13 × base area × height.
    13 × 36 × 10 = 120 cm³.

18. Question 18 · Composite shapes · G17

    A shape is a 10 cm by 6 cm rectangle with a semicircle of diameter 6 cm on one end. Find its area to 1 d.p.

    Answer:

    Rectangle 60 cm² and semicircle 12 × π × 3² = 14.14 cm².
    74.1 cm².

19. Question 19 · Arcs · G18

    Find the length of an arc of a circle with radius 9 cm and angle 80°, in terms of π.

    Answer:

    Arc = 80360 × 2π × 9.
    4π cm.

20. Question 20 · Sectors · G18

    Find the area of a sector with radius 6 cm and angle 120°, in terms of π.

    Answer:

    Area = 120360 × π × 36.
    12π cm².

21. Question 21 · Sectors · G18 · Higher only

    Find the perimeter of a sector with radius 10 cm and angle 72°, to 1 d.p.

    Answer:

    Arc = 72360 × 20π = 4π = 12.57 cm.
    Perimeter = 12.57 + 20 = 32.6 cm.

22. Question 22 · Similarity · G19

    What is the difference between congruent and similar shapes?

    Answer:

    Congruent shapes are identical in shape and size.
    Similar shapes have the same shape but are different sizes, with sides in proportion.

23. Question 23 · Similarity · G19

    Triangle A has sides 3, 4, 5. Triangle B has sides 6, 8, 10. Are they similar? Explain.

    Answer:

    Each side of B is 2 × the matching side of A.
    Yes; the sides are in the same ratio.

24. Question 24 · Pythagoras · G20

    Find the hypotenuse of a right-angled triangle with shorter sides 9 cm and 12 cm.

    Answer:

    c² = 9² + 12² = 81 + 144 = 225.
    c = 15 cm.

25. Question 25 · Pythagoras · G20

    The hypotenuse is 13 cm and one side is 5 cm. Find the other side.

    Answer:

    a² = 13² − 5² = 169 − 25 = 144.
    a = 12 cm.

26. Question 26 · Pythagoras · G20

    Find the distance between (1, 2) and (7, 10).

    Answer:

    Horizontal 6, vertical 8. d² = 36 + 64 = 100.
    d = 10.

27. Question 27 · Pythagoras · G20

    Is a triangle with sides 7, 24 and 25 right-angled?

    Answer:

    7² + 24² = 49 + 576 = 625 = 25².
    Yes.

28. Question 28 · Trigonometry · G20

    In a right-angled triangle, the hypotenuse is 10 cm and the angle is 30°. Find the opposite side.

    Answer:

    sin 30° = opp10.
    opp = 10 × 0.5 = 5 cm.

29. Question 29 · Trigonometry · G20

    Adjacent = 8 cm, hypotenuse = 10 cm. Find the angle to 1 d.p.

    Answer:

    cos θ = 810 = 0.8.
    θ = cos⁻¹ 0.8 = 36.9°.

30. Question 30 · Trigonometry · G20

    Opposite = 7 cm, adjacent = 4 cm. Find the angle to 1 d.p.

    Answer:

    tan θ = 74 = 1.75.
    θ = 60.3°.

31. Question 31 · Trigonometry · G20

    A ladder of length 6 m makes an angle of 70° with the ground. How high up the wall does it reach, to 1 d.p.?

    Answer:

    Height = 6 sin 70°.
    6 × 0.9397 = 5.6 m.

32. Question 32 · 3D Pythagoras · G20 · Higher only

    Find the length of the diagonal of a cuboid 3 cm by 4 cm by 12 cm.

    Answer:

    d² = 3² + 4² + 12² = 9 + 16 + 144 = 169.
    d = 13 cm.

33. Question 33 · 3D trigonometry · G20 · Higher only

    How do you find the angle between a line and a plane?

    Answer:

    Project the line onto the plane to form a right-angled triangle.
    Use trigonometry on that triangle.

34. Question 34 · Exact trig values · G21

    What is the exact value of tan 45°?

    Answer:

    An isosceles right-angled triangle has equal opposite and adjacent sides.
    tan 45° = 1.

35. Question 35 · Exact trig values · G21

    What is the exact value of sin 60°?

    Answer:

    From an equilateral triangle of side 2, halved.
    √32.

36. Question 36 · Exact trig values · G21

    What is the exact value of cos 60°?

    Answer:

    From an equilateral triangle of side 2, halved.
    12.

37. Question 37 · Exact trig values · G21

    What is the exact value of tan 30°?

    Answer:

    1√3.
    Rationalised: √33.

38. Question 38 · Exact trig values · G21

    State sin 0° and cos 0°.

    Answer:

    At 0°, the opposite side has zero length.
    sin 0° = 0 and cos 0° = 1.

39. Question 39 · Sine rule · G22 · Higher only

    State the sine rule.

    Answer:

    asin A = bsin B = csin C.
    Use it when you know a side and its opposite angle.

40. Question 40 · Cosine rule · G22 · Higher only

    State the cosine rule for finding side a.

    Answer:

    a² = b² + c² − 2bc cos A.
    Use it with two sides and the included angle.

41. Question 41 · Cosine rule · G22 · Higher only

    Find a when b = 5, c = 8 and A = 60°.

    Answer:

    a² = 25 + 64 − 2 × 5 × 8 × 0.5 = 89 − 40 = 49.
    a = 7.

42. Question 42 · Sine rule · G22 · Higher only

    In triangle ABC, a = 10, A = 30° and B = 45°. Find b to 1 d.p.

    Answer:

    bsin 45° = 10sin 30°, so b = 10 × 0.70710.5.
    b = 14.1.

43. Question 43 · Area of a triangle · G23 · Higher only

    Find the area of a triangle with sides 6 cm and 8 cm and included angle 30°.

    Answer:

    Area = 12ab sin C.
    0.5 × 6 × 8 × 0.5 = 12 cm².

44. Question 44 · Ambiguous case · G23 · Higher only

    Why can the sine rule give two possible angles?

    Answer:

    sin θ = sin(180° − θ).
    If the angle found is acute, the obtuse angle 180° − θ may also fit.

45. Question 45 · Area · G16

    A rectangle has area 54 cm² and length 9 cm. Find its width.

    Answer:

    Width = area ÷ length.
    6 cm.

46. Question 46 · Circles · G17

    Find the radius of a circle with area 154 cm², using π = 227.

    Answer:

    r² = 154 ÷ 227 = 49.
    r = 7 cm.

47. Question 47 · Volume · G16

    A cube has volume 125 cm³. Find the length of one side.

    Answer:

    ∛125.
    5 cm.

48. Question 48 · Frustums · G17 · Higher only

    How do you find the volume of a frustum of a cone?

    Answer:

    Find the volume of the whole cone.
    Subtract the volume of the small cone removed from the top.

49. Question 49 · Area · G16

    Find the area of a kite with diagonals 10 cm and 6 cm.

    Answer:

    The kite splits into two triangles, each with base 10 cm and total height 6 cm.
    Area = 12 × 10 × 6 = 30 cm².

50. Question 50 · Trigonometry · G20

    A ramp rises 1.5 m over a horizontal distance of 8 m. Find the angle of the ramp to 1 d.p.

    Answer:

    tan θ = 1.58 = 0.1875.
    θ = 10.6°.

More Edexcel GCSE Maths topics

• N1–N9 Structure and calculation
• N10–N12 Fractions, decimals and percentages
• N13–N16 Measures and accuracy
• A1–A7 Notation, vocabulary and manipulation
• A8–A16 Graphs
• A17–A22 Solving equations and inequalities
• A23–A25 Sequences
• R1–R16 Ratio, proportion and rates of change
• G1–G15 Properties and constructions
• G16–G23 Mensuration and calculation
• G24–G25 Vectors
• P1–P9 Probability
• S1–S6 Statistics

Other Maths courses: AQA GCSE Maths, OCR GCSE Maths.