A square is inscribed in a circle of radius r. Show that the ratio of the area of the square to the area of the circle is equal to 2π ≈ 0.637.

Step-by-step solution

Idea: A square’s diagonals are equal and meet at right angles, so its area is half the product of the diagonals; here each diagonal is a diameter.

1. The diagonals of a square are equal and bisect each other, so their meeting point is equidistant from all four corners: it is the centre O, and each diagonal is a diameter, 2r.1 mark
2. The diagonals are perpendicular, so area of square = 12 × 2r × 2r = 2r². (Or: side² + side² = (2r)² ⇒ side² = 2r².)1 mark
3. Ratio = 2r²πr² = 2π ≈ 23.1416 ≈ 0.637.1 mark

Check: Radius 1: square side √2, area 2; circle area π ≈ 3.14; 2 ÷ 3.14 ≈ 0.637 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Taking the side of the square as 2r (the diameter is the diagonal, not the side).
• Taking the side as r, giving r².

How this can come in the exam

Try one yourself

A square of area 50 cm² is inscribed in a circle. Find the radius of the circle.

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