Question #61bb3

1 Answer
Feb 17, 2018

The perimeter of a square inscribed in a circle with radius #r# is #4sqrt2r#.

Explanation:

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I will call the side length of the square #x#. When we draw in the diagonals of the square, we see that they form four right-angled triangles. The legs of the right angle triangles are the radius, and the hypotenuse is the side length of the square.

This means that we can solve for #x# using the Pythagorean Theorem:

#r^2+r^2=x^2#

#2r^2=x^2#

#sqrt(2r^2)=sqrt(x^2)#

#sqrt(2)sqrt(r^2)=x#

#x=sqrt2r#

The perimeter of the square is just the side length times four (all the side lengths are equal per definition of the square), so the perimeter is equal to:

#4x=4sqrt2r#