How do you prove that the area of a circle circumscribed about a square is twice the area…?

Ahaha… I’m really not good at geometry. I get basic stuff, but I really don’t get this once The full problem is "Prove that the area of a circle circumscribed about a square is twice the area of the circle inscribed within the square"

Let s = the length of the side of a square.
Then the area of a circle inscribed in the square = (pi s^2)/4; in this case s is the diameter of the circle.

The diagonal of the square is s sqrt(2)

A circle circumscribed about the square would have this diagonal as its diameter.

Its area would be [pi (s sqrt(2))^2]/4 = [pi x 2s^2]/4

This will be a lot easier to understand if you copy it out in standard math notation; wish we could use equation editor on here)

Good luck!

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2 Responses to “How do you prove that the area of a circle circumscribed about a square is twice the area…?”

Razz on March 11th, 2010 at 1:55 am

Let s = the length of the side of a square.
Then the area of a circle inscribed in the square = (pi s^2)/4; in this case s is the diameter of the circle.

The diagonal of the square is s sqrt(2)

A circle circumscribed about the square would have this diagonal as its diameter.

Its area would be [pi (s sqrt(2))^2]/4 = [pi x 2s^2]/4

This will be a lot easier to understand if you copy it out in standard math notation; wish we could use equation editor on here)

Good luck!
References :
C Dawg on March 11th, 2010 at 2:14 am

E.g. AOC=2 Square it and divide by 2 then add 2 due to AOC and you have proven it
References :

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