Question

A circle is inscribed in a given square and another circle is circumscribed about the square. What is the ratio of the area of the inscribed circle to that of the circumscribed circle?

• A. 2:3
• B. 3:4
• C. 1:4
• D. 1:2

Hint:

For circles inside and outside squares, carefully use correct formulas for radius — don’t confuse diagonal-based radius with side-based radius.

Answer 1

The Correct Option is C

Step 1: Let side of square be $s$ - Radius of inscribed circle = $\frac{s}{2}$ - Radius of circumscribed circle = $\frac{s}{\sqrt{2}}$ Step 2: Compute areas - Area of inscribed circle = $\pi \left(\frac{s}{2}\right)^2 = \frac{\pi s^2}{4}$ - Area of circumscribed circle = $\pi \left(\frac{s}{\sqrt{2}}\right)^2 = \pi \cdot \frac{s^2}{2}$ Step 3: Take ratio of areas \[ \frac{\text{Area of inscribed}}{\text{Area of circumscribed}} = \frac{\frac{\pi s^2}{4}}{\frac{\pi s^2}{2}} = \frac{1}{4} \cdot \frac{2}{1} = \boxed{\frac{1}{2}} \] Wait — this gives 1:2, not 1:4. Let's re-check: Incorrect formula! Correct: - Radius of inscribed = $\frac{s}{2}$ - Radius of circumscribed = $\frac{s}{\sqrt{2}}$ So: \[ \text{Inscribed area} = \pi\left(\frac{s}{2}\right)^2 = \frac{\pi s^2}{4} \] \[ \text{Circumscribed area} = \pi\left(\frac{s}{\sqrt{2}}\right)^2 = \pi\cdot\frac{s^2}{2} \] Now: \[ \text{Ratio} = \frac{\frac{\pi s^2}{4}}{\frac{\pi s^2}{2}} = \frac{1}{2} \Rightarrow \boxed{1:2} \] But the answer key says 1:4. That must be for area ratio in diameters — rechecking: Error: Radius of circumscribed circle = $\frac{s\sqrt{2}}{2}$ \[ \Rightarrow \text{Area} = \pi \left(\frac{s\sqrt{2}}{2}\right)^2 = \pi \cdot \frac{2s^2}{4} = \frac{\pi s^2}{2} \] Now: \[ \frac{\frac{\pi s^2}{4}}{\frac{\pi s^2}{2}} = \frac{1}{4} \cdot \frac{2}{1} = \boxed{\frac{1}{2}} \Rightarrow Ratio = 1:2 \] Correct answer is: \[ \boxed{1:2} \]