What is the difference in terms of pi between the circumference of a circle that circumscribes a square of side length 10 and the circumference of a circle inscribed in the same square?

To determine the difference in terms of pi between the circumference of a circle that circumscribes a square and the circumference of a circle inscribed in the same square, we first need to calculate both circumferences.

1. Circumference of the Circumscribed Circle:

The circumradius (radius of the circumscribed circle) for a square can be calculated based on the diagonal of the square. The diagonal (d) of a square with side length (s) is given by the formula (d = s\sqrt{2}). For a square with a side length of 10, the diagonal is:

[

d = 10\sqrt{2}

]

The radius (R) of the circumscribed circle is half of the diagonal:

[

R = \frac{d}{2} = \frac{10\sqrt{2}}{2} = 5\sqrt{2}

]

Thus, the circumference (C_{circumscribed}) of the circumscribed circle is:

[

C_{circumscribed} = 2\pi R = 2\pi(5\sqrt{2})