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Maths olympiad question · P6★★★★ Expert

Circles: olympiad question 10

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

A park lawn has a regular hexagonal pavilion with sides of 30 m, grass all round it.

A cow is tied by a 50 m rope to one corner of it (shown) and cannot go inside.

(Take π as 3.14.)

What area of grass can the cow reach, in m²?

Show the hints (3)
  1. Hint 1: How big is each angle of a regular hexagon? Round the corner it is tied to, what angle can the rope sweep?
  2. Hint 2: The rope is longer than a side: past each neighbouring corner 50 − 30 = 20 m is left to swing round. Through what angle?
  3. Hint 3: The big sector has radius 50 m and angle 360° − 120° = 240°; the two small ones radius 20 m and angle 180° − 120° = 60° each.

Worked solution

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

  1. 1.Each angle of a regular hexagon is (6 − 2) × 180° ÷ 6 = 120°. Tied at a corner, the pavilion blocks 120°, so the cow roams a sector of 360° − 120° = 240° with radius 50 m.
  2. 2.The big sector = 3.14 × 50 × 50 × 240/360 = 3.14 × 2500 × 2/3 m² (no need to work it out yet).
  3. 3.Along a side to the next corner, 50 − 30 = 20 m is left to swing round it: from the side's extension to the next side is 180° − 120° = 60°. There is one such small sector on each side.
  4. 4.The two small sectors = 2 × 3.14 × 20 × 20 × 60/360 = 3.14 × 400 × 1/3 m² (20 m is no longer than a side, so it never rounds a second corner).
  5. 5.Area = 3.14 × (2500 × 2 + 400) ÷ 3 = 3.14 × 5400 ÷ 3 = 3.14 × 1800 = 5652 m².
  6. ⚠️Common mistake: using three quarters of a circle (as for a square hut), or 90° for the small sectors. A hexagon's corner is 120°, so the big sector is 240° and each small one only 60°.

Answer: 5652 m²

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