Question

Find the area of a square inscribed in a circle of radius 4 cm.

• A. 32 cm\(^2\)
• B. 18 cm\(^2\)
• C. 64 cm\(^2\)
• D. 25 cm\(^2\)

Hint:

For a square inscribed in a circle, the diagonal equals the diameter of the circle.

Answer 1

The Correct Option is C

Step 1: Understand the problem.
A square is inscribed in a circle, meaning the circle passes through all four vertices of the square. The diagonal of the square equals the diameter of the circle. Step 2: Find the diagonal of the square.
Radius of the circle \(r = 4 \text{ cm}\)
Diameter \(d = 2r = 8 \text{ cm}\)
So, diagonal of square \(= 8 \text{ cm}\) Step 3: Use the relationship between side \(s\) and diagonal \(d\) of a square:
\[ d = s \sqrt{2} \] \[ s = \frac{d}{\sqrt{2}} = \frac{8}{\sqrt{2}} = 8 \times \frac{\sqrt{2}}{2} = 4\sqrt{2} \text{ cm} \] Step 4: Calculate the area of the square:
\[ \text{Area} = s^2 = (4\sqrt{2})^2 = 16 \times 2 = 32 \text{ cm}^2 \] Step 5: Notice a discrepancy: The highlighted correct answer in the image is 64 cm\(^2\), but the calculation shows 32 cm\(^2\). Double-check:
\[ s = \frac{8}{\sqrt{2}} = 4\sqrt{2} \approx 5.656 \] \[ \text{Area} = 5.656^2 \approx 32 \] So the mathematically correct answer is 32 cm\(^2\) (Option A), but the image shows (C) 64 cm\(^2\) as correct, possibly a mistake in the key.