OxfordAQA International GCSE Mathematics: Units, area, volume and surface area (9260) – 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 from your board or school.

These questions are for OxfordAQA International GCSE Mathematics (9260), specification Version 5.1 (for exams May/June 2018 onwards), section 3.3.2 Mensuration and calculation, references G14 to G16. Questions 9 to 12 are Extension only. Learn the content first in the study guide and the revision notes. See also the course hub, the printable checklist and the free 10-minute diagnostics.

Questions

1. (a) Convert 8.2 m³ to litres. [1] (b) Convert 450 cm² to m². [1] (c) Convert 90 km/h to m/s. [2]

2. A coach leaves at 20:50 and the journey takes 5 h 45 min. It travels 391 km. (a) Write the arrival time in 24 hour form. [2] (b) Calculate the average speed in km/h. [2] (c) Convert your answer to part (b) into m/s, to 1 decimal place. [1]

3. A metal block measures 9 cm by 4 cm by 2.5 cm and has mass 0.774 kg. (a) Calculate its density in g/cm³. [3] (b) Write its density in kg/m³. [1]

4. (a) A trapezium has parallel sides 9.2 cm and 14.8 cm, which are 7.5 cm apart. Find its area. [2] (b) A parallelogram has area 117 cm² and base 13 cm. Find its perpendicular height. [2]

5. A tank is a prism. Its cross-section is a trapezium with parallel sides 1.2 m and 2.0 m, which are 0.9 m apart. The tank is 3 m long. (a) Calculate the volume of the tank in m³. [3] (b) Write the capacity in litres. [1] (c) Water enters at 36 litres per minute. How many hours does it take to fill the tank? [2]

6. (a) A circle has diameter 14 cm. Write its circumference and area as multiples of π. [3] (b) A different circle has circumference 40 cm. Calculate its area to 3 significant figures. [3]

7. A rectangle is 14 cm by 8 cm. A semicircle of diameter 8 cm is cut out of one of the 8 cm ends, so that the shape has a curved notch. (a) Find the perimeter of the shape, in terms of π. [3] (b) Find the area of the shape, in terms of π. [3]

8. A circular running track has outer radius 25 m. The track is 4 m wide. (a) Find the area of the track, in terms of π. [3] (b) Surfacing costs £12 per m². Find the cost, to the nearest pound. [2]

9. (Extension only) A solid hemisphere has radius 6 cm. (a) Find its volume in terms of π. [2] (b) Find its total surface area in terms of π. [2]

10. (Extension only) A cone has base radius 6 cm and perpendicular height 8 cm. (a) Find the slant height. [1] (b) Find the volume, to 3 significant figures. [2] (c) Find the curved surface area, to 3 significant figures. [2]

11. (Extension only) A solid cone has height 24 cm and base radius 9 cm. A cut parallel to the base removes a smaller cone of height 10 cm from the top, leaving a frustum. (a) Find the radius of the cone that was removed. [2] (b) Find the volume of the frustum, to 3 significant figures. [4] (c) The frustum is made from a material of density 2.4 g/cm³. Find its mass in kg, to 3 significant figures. [2]

12. (Extension only) A solid is made from a cube of edge 10 cm with a square-based pyramid fixed on top. The base of the pyramid is the top face of the cube. The pyramid has perpendicular height 12 cm. (a) Find the volume of the solid. [3] (b) Find the total surface area of the solid. [4]

Answers

1. (a) 8.2 × 1000 = 8200 litres [1] (b) 450 ÷ 10 000 = 0.045 m² [1] (c) 90 × 1000 ÷ 3600 [1] = 25 m/s [1] Examiner insight: Write the conversion factor you are using. For an area, the factor is squared, so it is 100 × 100 rather than 100.

2. (a) 20:50 + 5 h 45 min = 26:35 [1], which is 02:35 [1] (b) 5 h 45 min = 5.75 h [1]; 391 ÷ 5.75 = 68 km/h [1] (c) 68 ÷ 3.6 = 18.9 m/s [1] Examiner insight: Turning 5 h 45 min into 5.45 h is a frequent slip. Dividing the minutes by 60 and showing it makes your method clear even if you slip later.

3. (a) Volume = 9 × 4 × 2.5 = 90 cm³ [1]; mass = 774 g [1]; density = 774 ÷ 90 = 8.6 g/cm³ [1] (b) 8.6 × 1000 = 8600 kg/m³ [1] Examiner insight: Convert the mass to grams before dividing. A density with mixed units, such as kg/cm³, is not what the question asked for.

4. (a) ½ × (9.2 + 14.8) × 7.5 [1] = 90 cm² [1] (b) Height = 117 ÷ 13 [1] = 9 cm [1] Examiner insight: Show the formula with numbers substituted before evaluating. For part (b), rearranging first (height = area ÷ base) avoids multiplying instead of dividing.

5. (a) Cross-section = ½ × (1.2 + 2.0) × 0.9 [1] = 1.44 m² [1]; V = 1.44 × 3 = 4.32 m³ [1] (b) 4.32 × 1000 = 4320 litres [1] (c) 4320 ÷ 36 = 120 minutes [1] = 2 hours [1] Examiner insight: The length of 3 m is the prism’s length, not part of the trapezium. Check the final unit: the question asks for hours, not minutes.

6. (a) Circumference = π × 14 = 14π cm [1]; radius = 7 [1]; area = 49π cm² [1] (b) 2πr = 40, so r = 20/π [1]; A = π × (20/π)² [1] = 400/π = 127 cm² [1] Examiner insight: Using the radius 7 rather than the diameter 14 in πr² is the usual error. In part (b), keep r as 20/π rather than a rounded decimal, then round only the final answer.

7. (a) Straight edges: 14 + 14 + 8 [1]; arc = ½ × π × 8 = 4π [1]; perimeter = 36 + 4π cm [1] (b) Rectangle = 14 × 8 = 112 [1]; semicircle = ½ × π × 4² = 8π [1]; area = 112 − 8π cm² [1] Examiner insight: The notch replaces one 8 cm edge, so the straight 8 cm edge is not part of the outside. Subtract in part (b) because the semicircle is removed.

8. (a) Inner radius = 25 − 4 = 21 m [1]; area = π × 25² − π × 21² [1] = 625π − 441π = 184π m² [1] (b) 184π × 12 = 2208π = 6936.6… [1]; cost = £6937 [1] Examiner insight: The width of the track is the difference between the radii, not an inner radius of its own. Use the exact value 184π in part (b) rather than a rounded area.

9. (a) V = ½ × (4/3) × π × 6³ [1] = 144π cm³ [1] (b) Curved = ½ × 4π × 6² = 72π, flat face = π × 6² = 36π [1]; total = 108π cm² [1] Examiner insight: A solid hemisphere has a flat circular face as well as a curved surface. Leaving it out is the most common way to lose marks on total surface area.

10. (a) l = √(6² + 8²) = √100 = 10 cm [1] (b) V = (1/3) × π × 6² × 8 [1] = 96π = 302 cm³ [1] (c) Curved area = π × 6 × 10 [1] = 60π = 188 cm² [1] Examiner insight: Volume uses the perpendicular height 8, whereas the curved area uses the slant height 10. Write down which height you are using for each formula.

11. (a) Scale factor = 10/24 [1]; radius = 9 × 10/24 = 3.75 cm [1] (b) Large cone = (1/3) × π × 9² × 24 = 648π [1]; small cone = (1/3) × π × 3.75² × 10 = 46.875π [1]; frustum = 601.125π [1] = 1890 cm³ [1] (c) Mass = 1888.49 × 2.4 = 4532.4 g [1] = 4.53 kg [1] Examiner insight: The removed cone is an enlargement scaled down by 10/24, so radius and height must be in the same ratio. Do not pair the radius 9 with the height 10.

12. (a) Cube = 10³ = 1000 [1]; pyramid = (1/3) × 100 × 12 = 400 [1]; total = 1400 cm³ [1] (b) Slant height s: s² = 12² + 5² [1], so s = 13 [1]; four triangles = 4 × ½ × 10 × 13 = 260 [1]; five cube faces = 500, so total = 760 cm² [1] Examiner insight: The pyramid covers the top face of the cube, so only five cube faces are on the outside. The slant height uses half the base, 5, not the full 10.

Where marks are usually lost

• Using the plain length factor on areas and volumes, instead of its square or cube.
• Writing minutes as a decimal of an hour without dividing by 60.
• Using the diameter in πr², or the radius in a formula that needs the diameter.
• Using the sloping side of a parallelogram or trapezium as the height.
• Giving a rounded decimal where an exact multiple of π was requested, or the reverse.
• Counting an internal join in a composite perimeter, or forgetting a curved edge.
• Rounding in the middle of a multi-step question.
• Omitting units, or giving cm where cm² or cm³ is needed.
• Mixing up the two heights of a cone: slant for curved area, perpendicular for volume.
• Counting a face that is hidden inside a composite solid.

Next steps

• Revise from the revision notes.
• Read the full explanations in the study guide.
• Tick off the topic on the printable checklist and visit the course hub.
• Try all free 10-minute diagnostics.
• Talk it through with a tutor: book a free trial class.

Official syllabus

OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1 (for exams May/June 2018 onwards), published by OxfordAQA. Section 3.3.2 Mensuration and calculation.