A medium question on circles for P5–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 big and small circles have the same centre.
A square has its four corners on the big circle, and its four sides just touch the small circle.
The square's area is 1600 cm².
(Take π as 3.14.)
What is the area of the ring between the two circles, in cm²?
Show the hints (2)
💡Hint 1: Draw the square's two diagonals. How is the big circle's radius related to them?
💡Hint 2: Ring area = 3.14 × (R × R − r × r). You don't need R and r themselves, only R × R and r × r.
Worked solution
Have a go first, then read the solution step by step.
1.Draw the two diagonals: they cut the square into 4 triangles, each with two short sides equal to the big radius R, so each has area R × R ÷ 2. The square = 4 × R × R ÷ 2 = 2 × R × R, so R × R = 1600 ÷ 2 = 800.
2.The square's side equals the small circle's diameter. Its midlines cut it into 4 small squares of side r, so r × r = 1600 ÷ 4 = 400.
3.Ring area = 3.14 × (R × R − r × r) = 3.14 × (800 − 400) = 3.14 × 400 = 1256 cm².
⚠️Common mistake: taking the big circle's radius as half the square's side. That is the small circle's radius; the big radius is half a diagonal.
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