What is the ratio of the areas of the incircle and circumcircle of a square?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(1: 2\)
The incircle of a square touches its sides, so its diameter is the side; the circumcircle passes through the corners, so its diameter is the diagonal. Areas of circles are in the ratio of the squares of their radii.
Radius of the incircle \(\displaystyle r=\frac{a}{2}\)
Radius of the circumcircle \(\displaystyle R=\frac{a\sqrt2}{2}=\frac{a}{\sqrt2}\)
Ratio of the areas \(\displaystyle =\frac{\pi r^2}{\pi R^2}=\frac{a^2/4}{a^2/2}=\frac{1}{2}\), that is, 1 : 2