Why can I not subtract the radii first for a ring area?

Area depends on the square of the radius, so square first: 3.14 × (R × R − r × r).

How do I find the perimeter of a running track?

Add the two straight sides; the two arcs together make one full circle, 3.14 × diameter.

How do I work back to the outer radius from a ring area?

Divide by 3.14, add the square of the inner radius, then take the square root.

P6 Maths Composite Shapes Paper 3: Challenge

2026-11-09

Tutor Prep Guide · P6 Maths

Ten original items, each with an answer and a one-line explanation.

Direct answers　Items 1-3 are basic, 4-6 intermediate and 7-9 challenge, with item 10 as the error review. Time yourself first, then check every answer and redo the wrong ones.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	20
Difficulty mix	Foundation 3 · Intermediate 3 · Challenge 3 · Error redo 1
Chapter	Chapter: P6 Maths Composite Shapes with Circles
Level	P6

02　Questions (10 items)

FoundationFind the area of a semicircle of radius 10 cm. (Take pi = 3.14)
FoundationFind the area of a quarter circle of radius 8 cm.
FoundationA semicircle has diameter 14 cm. Find its perimeter, including the diameter.
IntermediateA ring has outer radius 15 cm and inner radius 9 cm. Find its area.
IntermediateA square has side 20 cm and contains its largest inscribed circle. Find the area outside the circle.
IntermediateFind the perimeter of a quarter circle of radius 7 cm.
ChallengeA rectangle is 18 cm by 8 cm and a semicircle is joined to one long side, using that side as its diameter. Find the total area.
ChallengeA ring has area 263.76 cm² and inner radius 4 cm. Find the outer radius.
ChallengeA running track is a 40 m rectangle with a semicircle at each end; both semicircles have diameter 20 m. Find the length of one lap.
Error redoError review: a pupil finds the area of a ring with radii 15 cm and 9 cm as 3.14 × (15 − 9) × (15 − 9) = 113.04 cm². What is the correct area?

03　Answers and one-line notes

#	Answer	One-line note
1	157 cm²	3.14 × 10 × 10 ÷ 2 = 157 cm².
2	50.24 cm²	3.14 × 64 ÷ 4 = 50.24 cm².
3	35.98 cm	The arc is 3.14 × 14 ÷ 2 = 21.98 cm, and adding the 14 cm diameter gives 35.98 cm.
4	452.16 cm²	3.14 × (225 − 81) = 3.14 × 144 = 452.16 cm².
5	86 cm²	The square is 400 cm² and the circle is 314 cm², so 86 cm² remains.
6	24.99 cm	The arc is 2 × 3.14 × 7 ÷ 4 = 10.99 cm, and two radii add 14 cm, giving 24.99 cm.
7	271.17 cm²	The rectangle is 144 cm² and the semicircle is 127.17 cm², giving 271.17 cm² in total.
8	10 cm	263.76 ÷ 3.14 = 84, and 84 + 16 = 100, so the outer radius is 10 cm.
9	142.8 m	The two straight sides give 2 × 40 = 80 m, and the arcs make one circle of 3.14 × 20 = 62.8 m, so one lap is 142.8 m.
10	452.16 cm²	Square first, then subtract: 3.14 × (225 − 81) = 452.16 cm². Subtracting the radii before squaring loses area.

04　What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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