Question

The ratio of the area of the inscribed circle to the area of the circumscribed circle of an equilateral triangle is

• A. \( \frac{1}{8} \)
• B. \( \frac{1}{6} \)
• C. \( \frac{1}{4} \)
• D. \( \frac{1}{2} \)

Hint:

In an equilateral triangle, the ratio of the area of the inscribed circle to the area of the circumscribed circle is \( \frac{1}{4} \). This is derived from the geometric properties of the triangle and the circles.

Answer 1

The Correct Option is C

In an equilateral triangle, the relationship between the area of the inscribed circle and the area of the circumscribed circle is a classic problem. Let's break it down:

1. Let the side length of the equilateral triangle be \(a\).
2. The radius \(r_i\) of the inscribed circle (incircle) is given by: \[ r_i = \frac{a \sqrt{3}}{6} \] The area of the inscribed circle \(A_i\) is: \[ A_i = \pi r_i^2 = \pi \left( \frac{a \sqrt{3}}{6} \right)^2 = \frac{\pi a^2}{12} \] 3. The radius \(r_c\) of the circumscribed circle (circumcircle) is given by: \[ r_c = \frac{a}{\sqrt{3}} \] The area of the circumscribed circle \(A_c\) is: \[ A_c = \pi r_c^2 = \pi \left( \frac{a}{\sqrt{3}} \right)^2 = \frac{\pi a^2}{3} \] 4. The ratio of the area of the inscribed circle to the area of the circumscribed circle is: \[ \text{Ratio} = \frac{A_i}{A_c} = \frac{\frac{\pi a^2}{12}}{\frac{\pi a^2}{3}} = \frac{1}{4} \] Thus, the correct answer is (C).