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 rectangle is 12 cm long and 5 cm wide, and its diagonal is 13 cm.
A circle is drawn through its four corners, and a semicircle outwards on each of its four sides, leaving four lunes.
(Take π as 3.14.)
What is the total area of the four lunes (shaded), in cm²?
Show the hints (3)
💡Hint 1: A semicircle's area = 3.14 × diameter × diameter ÷ 8. To compare semicircles, just compare diameter × diameter.
💡Hint 2: The whole figure can be split two ways: one contains the four lunes, the other the rectangle.
💡Hint 3: 12 × 12 + 5 × 5 = 13 × 13: the semicircles on a long and a short side together equal a semicircle of diameter 13 cm, so the four small ones equal the whole circle. Take that equal part from both splits: what is left?
Worked solution
Have a go first, then read the solution step by step.
1.A semicircle's area = 3.14 × diameter × diameter ÷ 8, so comparing the small semicircles with the big circle comes down to diameter × diameter. The circle's diameter is the rectangle's diagonal, 13 cm.
2.12 × 12 + 5 × 5 = 144 + 25 = 169 = 13 × 13: the semicircles on a long side and a short side together = a semicircle of diameter 13. So the four small semicircles = two of those = the whole circle.
3.The whole figure = the rectangle + the four small semicircles; it is also = the circle + the four lunes.
4.Take the equal parts away from both: the four lunes = the rectangle = 12 × 5 = 60 cm².
⚠️Common mistake: thinking every semicircle must be worked out with 3.14 — the numbers get messy and slips creep in. The 3.14 cancels out completely; the answer is the straight-sided shape's area.
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