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Maths olympiad question · P6★★★★ Expert

Circles: olympiad question 11

An expert question on circles for P6, with a step-by-step solution, diagrams and the answer. A P6 topic. Shaded shapes made of circles and squares are common in the Hua Xia Cup's P6 qualifying round and finals.

The question

An L-shaped planter is an 80 cm by 50 cm rectangle with a 60 cm by 20 cm rectangle cut from one corner (shown).

A paper disc of radius 10 cm rolls once right round the outside of the planter, always touching it, back to where it started.

(Take π as 3.14.)

What area does the disc sweep (not counting the planter itself), in cm²?

Show the hints (3)
  1. Hint 1: Along each side the disc sweeps a strip 20 cm wide. What is the planter's perimeter? (Did cutting the corner change it?)
  2. Hint 2: At a corner that sticks out, the disc swings round: what shape does it sweep? And at the corner that goes in — do the two strips overlap?
  3. Hint 3: There are 5 outward corners, each sweeping a quarter circle of radius 20 cm; at the inward corner the two strips overlap in a 20 × 20 square.

Worked solution

Have a go first, then read the solution step by step.

  1. 1.Cutting the corner leaves the perimeter as the rectangle's: (80 + 50) × 2 = 260 cm. Along each side the disc sweeps a strip 20 cm wide (a diameter): 260 × 20 = 5200 cm² in all.
  2. 2.There are 5 outward corners: swinging round each, the disc sweeps a quarter circle of radius 20 cm; together 5 × 3.14 × 20 × 20 ÷ 4 = 1570 cm².
  3. 3.At the inward corner the disc touches both sides at once; the two strips overlap there in a 20 × 20 = 400 cm² square, counted twice, so take it off once.
  4. 4.Swept area = 5200 + 1570 − 400 = 6370 cm².
  5. ⚠️Common mistake: adding a quarter circle at all 6 corners (the inward one has none), or missing the overlap there; another is using radius 10 cm for the quarter circles — the disc's far edge is a whole diameter, 20 cm, from the corner.

Answer: 6370 cm²

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