OxfordAQA International GCSE Mathematics: Similar figures and arcs, sectors (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 cover OxfordAQA International GCSE Mathematics (9260), specification Version 5.1 (for exams May/June 2018 onwards), section 3.3.2, statements G17 and G18. Every question is Extension only. If you need the method first, read the study guide or the revision notes. Course pages: the OxfordAQA IGCSE Mathematics hub and the checklist. Want a quick check of your level? Try the free 10-minute diagnostics.

Give answers to 3 significant figures unless a question says otherwise, and keep the full value of π until the final line.

Questions

1. (Extension only) Two similar crates have matching lengths in the ratio 4 : 7. Write down the ratio of their surface areas and the ratio of their volumes. [2]

2. (Extension only) Two photo frames are similar. The smaller has area 96 cm² and width 12 cm. The larger has area 150 cm². Work out the width of the larger frame. [3]

3. (Extension only) Two tins are mathematically similar. Their heights are 8 cm and 14 cm. The larger tin holds 1715 cm³.

(a) Work out the volume of the smaller tin. [3] (b) The paper label on the smaller tin has area 88 cm². Work out the area of the label on the larger tin. [2]

4. (Extension only) A sector has radius 15 cm and angle 48°. Give each answer in terms of π.

(a) Find the arc length. [2] (b) Find the area of the sector. [2]

5. (Extension only) A sector has radius 12 cm and arc length 20 cm. Work out the angle of the sector, to 1 decimal place. [3]

6. (Extension only) A sector has angle 110° and area 55 cm².

(a) Work out the radius. [3] (b) Work out the perimeter of the sector. [3]

7. (Extension only) A model of a statue is 24 cm tall. The real statue is 3.6 m tall. The model is similar to the real statue.

(a) The surface area of the model is 1500 cm². Work out the surface area of the real statue, in m². [3] (b) The model holds 2400 cm³ of sand. Work out the volume of the real statue, in m³. [3]

8. (Extension only) Two vases are similar. The labels on them have areas 45 cm² and 80 cm². The smaller vase holds 405 ml. Work out the capacity of the larger vase. [4]

9. (Extension only) A garden bed is in the shape of a sector of a circle with radius 6.5 m and angle 140°.

(a) Work out its area. [2] (b) Edging is fitted all the way round the bed. Work out the total length of edging. [3]

10. (Extension only) A fan-shaped window frame is made by taking a sector of radius 20 cm and angle 72°, and removing from it a concentric sector of radius 8 cm and the same angle.

(a) Show that the area of the frame is 67.2π cm². [3] (b) Work out the total length of the frame’s outer boundary, which has two arcs and two straight edges. [4]

11. (Extension only) Sector P has radius 8 cm and angle 135°. Sector Q has the same angle and is similar to P. The radius of Q is 14 cm.

(a) Find the arc length of P, in terms of π. [2] (b) Write down the scale factor from P to Q and find the arc length of Q, in terms of π. [2] (c) Find the area of Q to 4 significant figures, using area P = 24π cm² in your working. [3]

12. (Extension only) A pendulum has length 85 cm and swings through an angle of 24° at its lowest point.

(a) Work out the length of the arc travelled by the end of the pendulum. [2] (b) On a different swing, the end travels along an arc of 50 cm. Work out the angle of the swing, to 1 decimal place. [2]

Answers

1. Area ratio = 4² : 7² = 16 : 49 [1]. Volume ratio = 4³ : 7³ = 64 : 343 [1]. [2] Examiner insight: write the squared and cubed numbers separately, because it is easy to give 8 : 14 for an area ratio by only doubling the lengths.

2. Area ratio = 150 : 96 = 25 : 16 [1]. Length ratio = √25 : √16 = 5 : 4, so k = 5/4 [1]. Width = 12 × 5/4 = 15 cm [1]. [3] Examiner insight: square-rooting the area ratio is the key step here. If you multiply 12 by the area scale factor instead, the working shows which idea you used.

3. (a) k = 14/8 = 7/4 [1]. Volume factor from small to large = (7/4)³ = 343/64 [1]. Small volume = 1715 × 64/343 = 320 cm³ [1]. (b) Area factor = (7/4)² = 49/16 [1]. Label = 88 × 49/16 = 269.5 cm² [1]. [5] Examiner insight: the larger figure is given here, so you must divide by the volume factor (or multiply by its reciprocal). A smaller tin with the bigger volume is a signal that something went the wrong way round.

4. (a) Fraction = 48/360 = 2/15 [1]. Arc = 2/15 × 2π × 15 = 4π cm [1]. (b) Area = 2/15 × π × 225 [1] = 30π cm² [1]. [4] Examiner insight: when the answer is wanted in terms of π, cancelling the fraction first and leaving π unevaluated gives a clean exact answer. A decimal can lose credit if the question asked for π.

5. Circumference = 2π × 12 = 24π [1]. θ = 20/(24π) × 360 [1] = 95.49… so 95.5° [1]. [3] Examiner insight: the fraction of the circle is arc ÷ circumference. Writing that fraction down before multiplying by 360 shows the method even if a keying slip changes the final value.

6. (a) 55 = 110/360 × π × r² [1]. r² = 55 × 360 / (110π) = 180/π = 57.29… [1]. r = 7.569… so 7.57 cm [1]. (b) Arc = 110/360 × 2π × 7.569… = 14.53… [1]. Perimeter = arc + 2r [1] = 14.53 + 15.14 = 29.7 cm [1]. [6] Examiner insight: keep the unrounded radius for part (b). Using 7.57 can nudge the third significant figure, and the perimeter needs both radii as well as the arc.

7. (a) Real height = 360 cm, so k = 360 ÷ 24 = 15 [1]. Area factor = 15² = 225 [1]. Real area = 1500 × 225 = 337 500 cm² = 33.75 m² [1]. (b) Volume factor = 15³ = 3375 [1]. Real volume = 2400 × 3375 = 8 100 000 cm³ [1] = 8.1 m³ [1]. [6] Examiner insight: the lengths are in different units, so convert before dividing. A final conversion from cm² to m² uses 10 000, and from cm³ to m³ uses 1 000 000.

8. Area ratio = 45 : 80 = 9 : 16 [1]. Length ratio = 3 : 4, so k = 4/3 [1]. Volume factor = 64/27 [1]. Capacity = 405 × 64/27 = 960 ml [1]. [4] Examiner insight: this question needs two changes of power in one chain: area to length, then length to volume. Writing each ratio on its own line makes the route clear.

9. (a) Area = 140/360 × π × 6.5² [1] = 51.6 m² [1]. (b) Arc = 140/360 × 2π × 6.5 = 15.88… [1]. Straight edges = 2 × 6.5 = 13 [1]. Total = 15.88 + 13 = 28.9 m [1]. [5] Examiner insight: “all the way round” signals the perimeter. The arc alone is a common answer, so state clearly that you are adding the two radii.

10. (a) Area = 72/360 × π × (20² − 8²) [1]. = 1/5 × π × 336 [1] = 67.2π, as required [1]. (b) Outer arc = 72/360 × 2π × 20 = 8π [1]. Inner arc = 72/360 × 2π × 8 = 3.2π [1]. Straight edges = 2 × (20 − 8) = 24 [1]. Total = 11.2π + 24 = 59.2 cm [1]. [7] Examiner insight: on a “show that” question, the line with the exact result must appear with the working that leads to it. A bare statement of 67.2π is unlikely to earn the final step.

11. (a) Fraction = 135/360 = 3/8 [1]. Arc P = 3/8 × 16π = 6π cm [1]. (b) k = 14/8 = 1.75 [1]. Arc Q = 6π × 1.75 = 10.5π cm [1]. (c) Area factor = 1.75² = 3.0625 [1]. Area Q = 24π × 3.0625 = 73.5π [1] = 230.9 cm² (4 s.f.) [1]. [7] Examiner insight: sectors with the same angle are similar, so scale the arc by k and the area by k². You can check the answer by working out area Q directly from 3/8 × π × 14².

12. (a) Arc = 24/360 × 2π × 85 [1] = 35.6 cm [1]. (b) θ = 50/(2π × 85) × 360 [1] = 33.70… so 33.7° [1]. [4] Examiner insight: the radius is the pendulum length, not the diameter. Check that the angle in part (b) is bigger than in part (a) when the arc is longer.

Where marks are usually lost

• Scaling an area by k instead of k², or a volume by k instead of k³.
• Forgetting to take a square root or cube root when the question gives areas or volumes and asks for a length.
• Dividing the wrong way round, so a larger figure ends up with a smaller answer.
• Leaving the two radii out of the perimeter of a sector.
• Using the diameter in a sector formula when the formula needs the radius.
• Rounding the radius or the arc early and then losing accuracy in the final answer.
• Giving a decimal when the question asks for an answer in terms of π, or the other way round.
• Forgetting to convert cm² to m² or cm³ to m³, or using 100 or 1000 as the conversion.
• Writing only a final answer on a multi-step or “show that” question, so the method cannot be seen.

Next steps

• Re-read the revision notes for the formulas and short reminders.
• Go back to the study guide for any step that felt unfamiliar.
• Visit the OxfordAQA IGCSE Mathematics hub and tick off statements on the checklist.
• Take the free 10-minute diagnostics to find your next weak spot.
• Book a free trial class if you would like a teacher to go through these questions with you.

Official syllabus

OxfordAQA International GCSE Mathematics (9260), specification Version 5.1 (for exams May/June 2018 onwards), published by OxfordAQA. Statements G17 and G18, section 3.3.2 Mensuration and calculation.