Question

In the following diagram, square ABCD is inscribed in a circle. The length of AD is 6 inches. What is the area of the circle in square inches?

Answer 1

as you can see on the following draw:

each triangle is right and isosceles. so we can use Pythagoras and find out the length of each side of the triangle, which is the radius of the circle:

$\begin{gathered} h^2=r^2+r^2 \\ h=6 \\ 6^2=r^2+r^2 \\ 36=2r^2 \\ \frac{36}{2}=r^2 \\ 18=r^2 \\ r=\sqrt[]{18}\approx4.24 \end{gathered}$

and the area of the circle is:

$\begin{gathered} \pi r^2 \\ \pi\cdot(4.24)^2 \\ \pi\cdot18=56.55in^2 \end{gathered}$

so the answer is: 18π square inches