Mathematics • Year 8 • Unit 3 • Lesson 7
Circumference — Mixed Challenge
Pull everything from Lesson 7 together: forward calculations, exact π form, reverse rearrangements and semicircle perimeters. Six mixed problems, one "find the mistake", and one open-ended π estimation challenge.
1. Mixed problems — choose the right move
Each question pulls a different idea from Lesson 7. Decide which formula and which form (exact π or decimal) you need before you write. 3 marks each
1.1 A circle has r = 12 cm. Find C in exact form (n π).
1.2 A circle has d = 18 m. Find C to 2 d.p.
1.3 A circle has C = 31.4 cm. Find r to the nearest cm (use π ≈ 3.14).
1.4 A semicircle has diameter 20 cm. Find its perimeter (curved arc + straight edge) to 1 d.p.
1.5 A bicycle wheel has radius 28 cm. Use π ≈ 22/7 to find exactly how many cm the wheel travels in 10 revolutions.
1.6 A quarter-circle (90° sector) has radius 6 cm. Find the total perimeter (curved arc + two straight radii) to 2 d.p.
2. Find the mistake
A Year 8 student has tried to find the circumference of a circle with diameter d = 14 cm using π ≈ 22/7. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — find C when d = 14 cm:
Line 1: C = 2πd
Line 2: C = 2 × (22/7) × 14
Line 3: C = 2 × 22 × 2 = 88
Line 4: So the circumference is 88 cm.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? Revisit lesson § Card 1 — when diameter is given, the formula is C = πd, not 2πd. The 2 only appears when using radius (C = 2πr).3. Open-ended challenge — measure π yourself
This question has more than one valid answer. 4 marks
3.1 Pick three different circular objects around your home (e.g. a plate, a coin, the rim of a cup, a clock face). For each object:
(i) Measure the diameter d (use a ruler — straight line across the centre) in cm.
(ii) Measure the circumference C by wrapping string around the edge, then measuring the string length, in cm.
(iii) Calculate the ratio C ÷ d.
(iv) Record your results in a small table (object | d | C | C ÷ d).
Then answer:
(a) What value should C ÷ d be close to for every circle, and why?
(b) Find the average of your three C ÷ d values. How close is it to π ≈ 3.14159?
(c) Suggest one reason your measured ratios were not exactly π.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — r = 12 (exact)
C = 2π × 12 = 24π cm (exact). (≈ 75.40 cm.)
1.2 — d = 18 m
C = π × 18 = 18π ≈ 56.55 m (2 d.p.).
1.3 — Find r from C = 31.4
r = C ÷ (2π) = 31.4 ÷ (2 × 3.14) = 31.4 ÷ 6.28 = 5 cm exactly. (To the nearest cm: 5 cm.)
1.4 — Semicircle perimeter (d = 20)
Arc = ½ × C = ½ × π × 20 = 10π ≈ 31.4 cm. Straight edge = d = 20 cm. Total perimeter = 31.4 + 20 = 51.4 cm (1 d.p.).
1.5 — Bicycle wheel r = 28, 10 revs
1 rev = C = 2 × (22/7) × 28 = 2 × 22 × 4 = 176 cm. 10 revs = 10 × 176 = 1 760 cm = 17.6 m exactly.
1.6 — Quarter-circle r = 6
Arc = ¼ × 2π × 6 = 3π ≈ 9.42 cm. Two straight radii = 2 × 6 = 12 cm. Total = 9.42 + 12 ≈ 21.42 cm.
2 — Find the mistake
(a) The mistake is on Line 1 (carried into all later lines).
(b) The formula C = 2πd is wrong — when diameter is given, the correct formula is C = πd. The factor of 2 belongs in C = 2πr (radius version) because d = 2r already builds in the doubling.
(c) Corrected working:
C = πd = (22/7) × 14 = 22 × 2 = 44 cm. ✓
Sanity check: the student's answer (88) is exactly double the correct answer — that's the giveaway that they doubled by mistake.
3 — Measure π (sample response)
There is no single correct numeric answer — student measurements will vary. A typical good response:
Sample table:
Plate: d = 24.0 cm, C = 75.5 cm, C ÷ d = 3.146
Mug rim: d = 8.5 cm, C = 26.8 cm, C ÷ d = 3.153
Clock face: d = 22.0 cm, C = 68.9 cm, C ÷ d = 3.132
(a) The ratio should always be close to π (≈ 3.14159) because π is defined as C ÷ d for any circle.
(b) Average of the three sample ratios = (3.146 + 3.153 + 3.132) ÷ 3 ≈ 3.144 — very close to π (off by about 0.003).
(c) Reasons measurements are not exact: string slips slightly; ruler reading has measurement error (typically ±1 mm); the object may not be perfectly circular; you may have measured a chord, not the true diameter.
Marking: 1 mark for three objects measured with table; 1 mark for ratios calculated; 1 mark for stating C ÷ d ≈ π with explanation; 1 mark for at least one valid source of measurement error.