Question

Find the ratio of the areas of the incircle and circumcircle of a square.

1:1

1:2

1:3

1:4

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Answer


The correct option is B: 1:2

Let the side of the square be $x$.

  1. The diagonal of the square is given by: $$ \text{Diagonal} = \sqrt{2} \cdot x $$

  2. The radius of the incircle: $$ r_{\text{incircle}} = \frac{x}{2} $$

  3. In the isosceles right triangle formed by half the diagonal as the hypotenuse, the radius of the circumcircle: $$ r_{\text{circumcircle}} = \frac{\sqrt{2} \cdot x}{2} = \frac{x}{\sqrt{2}} $$

  4. Finding the areas of the circles:

    • Area of the incircle: $$ A_{\text{incircle}} = \pi \left( \frac{x}{2} \right)^2 = \frac{\pi x^2}{4} $$
    • Area of the circumcircle: $$ A_{\text{circumcircle}} = \pi \left( \frac{x}{\sqrt{2}} \right)^2 = \pi \cdot \frac{x^2}{2} $$
  5. The required ratio of the areas is: $$ \text{Ratio} = \frac{\frac{\pi x^2}{4}}{\frac{\pi x^2}{2}} = \frac{1}{4} : \frac{1}{2} = 1 : 2

Thus, the ratio of the areas of the incircle and the circumcircle of a square is 1:2.


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