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, two squares of side 5 cm are put together.
Each has a quarter-circle arc inside it, centred on one of its 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: Are the quarter circles in the squares all the same size?
💡Hint 2: Shaded pieces can be cut out and moved. Try moving a shaded quarter circle into a square whose quarter circle is blank.
💡Hint 3: The 2 quarter circles are the same size, and as many are shaded as blank. After the move, how many whole squares does the shading fill? Each is 5 × 5 cm².
Worked solution
Have a go first, then read the solution step by step.
1.Every arc is centred on a corner of its square with the side, 5 cm, as radius, so both quarter circles are exactly the same size.
2.Cut out each shaded quarter circle, turn it, and move it into a square whose quarter circle is blank: it fills the gap exactly.
3.After the move, the shaded part is exactly 1 whole square: 5 × 5 = 25 cm².
⚠️Common mistake: working out each piece with 3.14 — every piece has decimals, and slips creep in. The shaded parts fit together exactly, so the 3.14 is never needed.
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