Outline Mathematics
Geometry

Circle in a Square Inscribed in a Circle

A square is inscribed in a circle and a second circle is inscribed in the square. What is the relationship between the areas of the two circles?

Solution

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Copyright © 1996-2017 Alexander Bogomolny

A square is inscribed in a circle and a second circle is inscribed in the square. What is the relationship between the areas of the two circles?

The problem is easily solved algebraically if one observes that the bigger circle has as a diameter the diagonal of the square while the small circle's diameter equals the side of the square. Since the diagonal of the square is 2 of the side and the areas of the circles relate as the squares of their diameters, the area of the big circle is twice that of the small one.

However, this one is very much related to the problem in which circle is inscribed in a square,square,circle,rhombus and a second circle,square,circle,rhombus is inscribed in the square,square,circle,rhombus.

The solution there exploits a charming argument by Socrates,Aristotle,Socrates,Kant,Apollonius. Here, too, we can readily perform the same feat.

Circumscribe a square around the big circle, with the sides at 45° angles to the sides of the given square. By Socrates' argument, the areas of the two squares are in the ratio 2:1. Their corresponding parts - the inscribed circles in particular - are in the same ratio.

(This is #7 from Archimedes' Book of Lemmas.)

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Copyright © 1996-2017 Alexander Bogomolny

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