+0

# Square

0
254
1

What is the ratio of the area of a square inscribed in a circle to the square circumscribing the circle?

Guest Dec 28, 2014

#1
+26399
+5

Let the radius of the circle be r.

The diameter of the circle is then 2r, and this is the length of the diagonal of the inscribed square.  This means the length of a side of the inscribed square is 2r/√2, so the area of the inscribed square is (2r/√2)2 = 2r2.

The diameter is also the length of a side of the circumscribed square, so the area of the circumscribed square is (2r)2 = 4r2.

Hence the ratio of the area of the inscribed square to that of the circumscribed square is 2r2/4r2 = 1/2

.

Alan  Dec 28, 2014
Sort:

#1
+26399
+5

Let the radius of the circle be r.

The diameter of the circle is then 2r, and this is the length of the diagonal of the inscribed square.  This means the length of a side of the inscribed square is 2r/√2, so the area of the inscribed square is (2r/√2)2 = 2r2.

The diameter is also the length of a side of the circumscribed square, so the area of the circumscribed square is (2r)2 = 4r2.

Hence the ratio of the area of the inscribed square to that of the circumscribed square is 2r2/4r2 = 1/2

.

Alan  Dec 28, 2014

### 30 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