Question

Question #61bb3

Answer 1

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

Explanation:

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`