Chapter 3, Problem 2E

To determine

To calculate: The exact ratio of the area of the circle with area of the square.

Answer to Problem 2E

The ratio of the area of the circle with area of the square is $\frac{\pi }{2}:1$ .

Explanation of Solution

Given Information:

Radius = r

Calculation:

Let $x$ be the side of the square.

Circle is inscribed in the square.

So radius $r=\frac{x}{\sqrt{2}}$ .

Area of the circle is $\pi r^2$ .

Area of the square is $x^2$ .

Find the ratio of the area of the circle with area of the square.

  $\begin{array}{c}\pi r^2:x^2\\ \pi {\left(\frac{x}{\sqrt{2}}\right)}^2:x^2\\ \left(\frac{\pi x^2}{2}\right):x^2\\ \frac{\pi }{2}:1\end{array}$

Hence, the ratio of the area of the circle with area of the square is $\frac{\pi }{2}:1$ .