+0

# What is the ratio of the area of the smaller circle to the area of the larger square?

0
66
1
+62

A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?

bbelt711  Aug 17, 2017

#1
+4476
+2

Let's call the side length of the larger square  " s "  .

Then...

diameter of the larger circle  =  s

Let's draw a diameter of the larger circle from the corner of the smaller square to the opposite corner of the smaller square. This creates a 45 - 45 - 90 triangle, where the side across from the 90° angle is  "s"  . So....

side across from the  45°  angle  =   $$\frac{s}{\sqrt2}$$

diameter of the smaller circle       =  $$\frac{s}{\sqrt2}$$

radius of the smaller circle            =  $$\frac12\,*\,\text{diameter}=\frac12\,*\,\frac{s}{\sqrt2}=\frac{s}{2\sqrt2}$$

$$\frac{\text{area of smaller circle}}{\text{area of larger square}}=\frac{\pi(\frac{s}{2\sqrt2})^2}{s^2}=\frac{\pi(\frac{s^2}{4*2})}{s^2}=\frac{\pi s^2}{8}\,*\,\frac1{s^2}=\frac{\pi}{8}$$

hectictar  Aug 17, 2017
edited by hectictar  Aug 17, 2017
edited by hectictar  Aug 17, 2017
Sort:

#1
+4476
+2

Let's call the side length of the larger square  " s "  .

Then...

diameter of the larger circle  =  s

Let's draw a diameter of the larger circle from the corner of the smaller square to the opposite corner of the smaller square. This creates a 45 - 45 - 90 triangle, where the side across from the 90° angle is  "s"  . So....

side across from the  45°  angle  =   $$\frac{s}{\sqrt2}$$

diameter of the smaller circle       =  $$\frac{s}{\sqrt2}$$

radius of the smaller circle            =  $$\frac12\,*\,\text{diameter}=\frac12\,*\,\frac{s}{\sqrt2}=\frac{s}{2\sqrt2}$$

$$\frac{\text{area of smaller circle}}{\text{area of larger square}}=\frac{\pi(\frac{s}{2\sqrt2})^2}{s^2}=\frac{\pi(\frac{s^2}{4*2})}{s^2}=\frac{\pi s^2}{8}\,*\,\frac1{s^2}=\frac{\pi}{8}$$

hectictar  Aug 17, 2017
edited by hectictar  Aug 17, 2017
edited by hectictar  Aug 17, 2017

### 18 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details