What is the ratio of the area of a circumscribed circle to an inscribed circle for the same square?

By determining the radius of the small inscribed circle (x/2) and the radius of the large circumscribed circle (x√2 / 2) in terms of the square's side length x, their respective areas are calculated as πx²/4 and πx²/2. The ratio of their areas A1 / A2 is exactly 2.

Two Circles and a Square – Area Ratio Problem & Solution

A problem, with a detailed solution, on a square inscribed in one circle and circumscribed around another is presented.

Problem Statement

The square \(ABCD\) is inscribed inside the larger circle \(C_1\) and the smaller circle \(C_2\) is inscribed inside the same square. If \(A_1\) is the area of the large circle and \(A_2\) is the area of the small circle, what is the ratio \( \dfrac{A_1}{A_2} \)?

two circles and one square - problem

Solution to Problem

Step 1: Area of the Smaller Circle (\(A_2\))

If \( x \) is the size of one side of the square, the small circle has a diameter of size \( d = x \) and its radius is \( \frac{x}{2} \). Hence its area \( A_2 \) is given by:

\[ A_2 = \pi \left(\frac{x}{2}\right)^2 = \frac{\pi x^2}{4} \]

Step 2: Radius of the Larger Circle (\(R\))

The radius \( R \) of the large circle is equal to \( \frac{D}{2} \), where \( D \) is the diagonal of the square. By Pythagoras' theorem:

\[ D^2 = x^2 + x^2 = 2x^2 \] \[ D = x \sqrt{2} \] \[ R = \frac{x \sqrt{2}}{2} \]

Step 3: Area of the Larger Circle (\(A_1\))

The area \( A_1 \) of the large circle is given by:

\[ A_1 = \pi \left(\frac{x \sqrt{2}}{2}\right)^2 = \frac{\pi x^2}{2} \]

Step 4: Calculating the Ratio of Areas

Hence, the ratio of the areas is:

\[ \frac{A_1}{A_2} = \frac{\frac{\pi x^2}{2}}{\frac{\pi x^2}{4}} = 2 \]

Two Circles and a Square – Problem With Solution

Problem Statement

Solution to Problem

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