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
In the figure, the square's side is 30 cm.
Two quarter-circle arcs are drawn inside it, centred on two opposite corners, with the side as radius.
(Take π as 3.14.)
What is the area of the shaded part, in cm²?
Show the hints (3)
💡Hint 1: How big are the two quarter circles together? Do they overlap?
💡Hint 2: Together the two quarter circles cover the square. Which part is covered twice?
💡Hint 3: The two quarter circles total 3.14 × 30 × 30 ÷ 2 = 1413 cm²; what they have beyond the square is the leaf, covered twice.
Worked solution
Have a go first, then read the solution step by step.
1.The twoquarter circles of radius 30 cm make half a circle: 3.14 × 30 × 30 ÷ 2 = 1413 cm².
2.Together they cover the whole square, and the leaf is covered twice.
3.So the leaf = two quarter circles − the square = 1413 − 900 = 513 cm².
⚠️Common mistake: taking one quarter circle less half the square and stopping — that is only half the leaf (one segment); it needs doubling.
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