Figuring the largest square to fit inside a circle (Footing Issue)

The argument requires the Pythagorean Theorem. Draw a circle with a square, as large as possible, inside the circle.

By the symmetry of the diagram the center of the circle is on the diagonal AB of the square. The length of AB is 60 inches and the lengths of BC and CA are equal. The Pythagorean Theorem then says that

|BC|² + |CA|² = |AB|²

Hence

2 |BC|² = 24² = 576

and therefore

|BC|² = 288

Taking the square root on my calculator I get

|BC| = 16.97

The largest square plate we can put in a 24 inch diameter circle is 16.97" inches on each side.
That's a little bigger than 16 and 15/16" (16.9375")