A hard 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 hoop of radius 25 cm rolls without slipping round the outside of a rectangular flower bed 294 cm long and 175 cm wide, keeping against its edge, once round and back to the start.
(Take π as 3.14.)
(Not to scale.)
How far does the hoop's centre travel altogether, in cm?
Show the hints (3)
💡Hint 1: Rolling along one side, how far does the hoop's centre move?
💡Hint 2: At each corner, the centre swings round an arc. What do the four corners' arcs make together?
💡Hint 3: Along the four sides the centre moves as far as the bed's perimeter; the four corner arcs make one whole circle of radius 25 cm: 2 × 3.14 × 25 = 157 cm.
Worked solution
Have a go first, then read the solution step by step.
1.Along each side, the centre moves exactly the side's length: all four sides make (294 + 175) × 2 = 938 cm.
2.At each corner, the centre swings round a quarter circle of radius 25 cm; the four corners make exactly one whole circle: 2 × 3.14 × 25 = 157 cm.
3.Altogether the centre travels 938 + 157 = 1095 cm.
⚠️Common mistake: taking the centre's path as the bed's perimeter, 938 cm, and missing the arcs at the four corners.
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