Mathematics • Year 8 • Unit 3 • Lesson 7
Circumference in the Real World
Use C = πd and C = 2πr where they actually show up: velodromes, bicycle wheels, clock hands, garden hoses and running tracks. Then explain your thinking in your own words.
1. Word problems
Each problem hides a circle. Identify whether you're given r or d, choose the right formula, then calculate. Use π ≈ 3.14 (or the π button) unless told otherwise.
1.1 — Velodrome laps. A circular velodrome has diameter 90 m. A cyclist needs to cover exactly 1 km in training.
(a) Find the length of one lap (the circumference) to 2 d.p.
(b) How many full laps does the cyclist need to complete?
(c) After 5 full laps, how much further (in m) until they reach 1 km? 3 marks
1.2 — Bicycle wheel. A bicycle wheel has radius 35 cm.
(a) How far does the wheel travel in one full revolution (to 2 d.p.)?
(b) How many revolutions are needed to travel 2.2 km? (Convert km → cm first.) 3 marks
1.3 — Clock hand sweep. A clock has a minute hand 15 cm long (from centre to tip) and an hour hand 10 cm long.
(a) How far does the tip of the minute hand travel in 1 hour?
(b) How far does the tip of the hour hand travel in 12 hours? 2 marks
1.4 — Garden hose loop. A coiled garden hose forms a circle of circumference 9.42 m.
(a) Find the diameter of the loop (use π ≈ 3.14, give to 1 d.p.).
(b) Find the radius. 2 marks
1.5 — Running track. A standard running track has two straight sections of 80 m each, and two semicircular ends of diameter 56 m.
(a) The two semicircles together form one full circle. Find the total length of the two curved ends.
(b) Find the total length of one lap (straights + curves) to 2 d.p. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate is asked to find the circumference of a circle with radius 7 cm. They write "C = π × 7 ≈ 22 cm". In your own words, explain (i) which part of the formula they forgot, (ii) what the correct answer should be (show working with π ≈ 22/7 for clean numbers), and (iii) one quick "sanity check" using diameter that would have warned them. Use the phrase "do not forget the factor of 2" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Velodrome laps
(a) One lap C = π × 90 ≈ 282.74 m.
(b) 1000 ÷ 282.74 ≈ 3.54, so 3 full laps (then more).
(c) After 5 full laps: 5 × 282.74 = 1413.7 m, which already exceeds 1 km. So after 3 full laps the cyclist has 1000 − (3 × 282.74) = 1000 − 848.22 = 151.78 m still to go to reach 1 km. (Sample assumes they meant "after enough laps to come close" — clearly 5 laps overshoots.)
1.2 — Bicycle wheel
(a) C = 2π × 35 = 70π ≈ 219.91 cm per revolution.
(b) 2.2 km = 220 000 cm. Revolutions = 220 000 ÷ 219.91 ≈ 1000.41 revolutions (~1000 rev).
1.3 — Clock hands
(a) Minute hand traces a circle of radius 15 in 1 hour. C = 2π × 15 = 30π ≈ 94.25 cm.
(b) In 12 hours the hour hand makes one full revolution: C = 2π × 10 = 20π ≈ 62.83 cm.
1.4 — Garden hose
(a) d = C ÷ π = 9.42 ÷ 3.14 = 3.0 m.
(b) r = d ÷ 2 = 1.5 m.
1.5 — Running track
(a) Two semicircles = one full circle: curved total = π × 56 ≈ 175.93 m.
(b) Lap total = 2 × 80 + 175.93 = 160 + 175.93 ≈ 335.93 m.
2.1 — Explain your thinking (sample response)
The classmate has used C = πr instead of C = 2πr — they have used the radius but forgotten to double it first. The formula is C = 2πr because the diameter is 2 × radius, so you must do not forget the factor of 2. Using π ≈ 22/7, the correct answer is C = 2 × (22/7) × 7 = 2 × 22 = 44 cm, exactly double their answer of 22. A quick sanity check: the diameter is 2r = 14 cm, and the circumference is always a little more than 3 times the diameter (because π ≈ 3.14), so the answer should be a bit more than 3 × 14 = 42 cm — and 44 fits this estimate, while 22 cm is way too small.
Marking: 1 mark for spotting the missing factor of 2; 1 mark for correct answer 44 cm with working; 1 mark for the sanity check (~3 × diameter); 1 mark for clear full-sentence explanation using "do not forget the factor of 2".