A square is inscribed in a circle such that each corner touches the circle

Question

A. Find a function that gives the area of the square as a a function of the radius of the circle. Simplify the function as much as possible, describe any symmetry in the function. Make certain to show all the steps you use to come up with the function, as well as describe your reasoning.

B. State the domain and range of the function you came up with in part (A) keeping in mind all practical considerations.

Answer 1

First, recognize that the diagonal of the square is also the diameter of the circle.

From the pythagorean theorem, we know that each side of the square is the diagonal divided by the square root of 2. side = (1 / √2)*diam

The area of the square is the square of a side: A = side²

And the diameter of the circle is twice the radius: diam = 2*radius

SO: A = side² = ( (1 / √2)*diam )² = ( (1 / √2)*( 2*radius ) )²

Area of the square is the square of 2 times the radius of the circle

A shorter way to get the same answer:

Each side of the square is the hypotenuse of a right triangle, where the 2 triangle sides are the radius of the circle.

The hypotenuse is the side times √2, So the ares is 2 times the square of the radius of the circle.

Draw a picture on graph paper to see all this