What is the difference in square centimeters between the area of a circle and the area of an inscribed square if the circle's diameter is 16 cm?

To determine the difference in area between a circle and an inscribed square when the circle's diameter is 16 cm, we start by calculating the radius of the circle. The radius is half of the diameter, which gives us a radius of 8 cm.

Next, we calculate the area of the circle using the formula for the area of a circle, ( A = \pi r^2 ). Substituting the radius:

[

A_{circle} = \pi (8)^2 = 64\pi , \text{cm}^2.

]

Now we need to find the area of the inscribed square. The diameter of the circle is also the diagonal of the square. If we denote the side length of the square as ( s ), we can relate the side length to the diagonal using the formula for the diagonal of a square, ( d = s\sqrt{2} ). Setting this equal to the circle's diameter:

[

s\sqrt{2} = 16 \implies s = \frac{16}{\sqrt{2}} = 8\sqrt{2} , \text{cm}.

]

Now, we can calculate the area of the square: