Question

Find the area of the shaded portion in the square.
(assuming the central point of each arc is the corresponding corner)

Answer 1

All you need to know to solve this problem are an understanding of how to find the area of a square and a circle. If you look at one of the shaded areas and imagine the rest of the square to be unshaded, it looks like a quarter of a circle. So you just need to find the area of that quarter of a circle, subtract it from the total area of the square, and multiply that by two to get the area of the other shaded area.
So first, how do you find the area of the top shaded half? Like we said, the rest of the square looks like a quarter of a circle. So we just need to find the area of a circle with a radius of 2 and divide that by 4 to get a quarter of the area.
\[\pi r^2\]\[\pi 2^2 = 4\pi\]\[\frac{4\pi}{4}=\pi\] So the area of a quarter of the circle is pi. Now you can subtract the area of a quarter of the circle (which we know to be pi) from the total area of the square (which we know is 4 sq. units). This gives you only one of the shaded areas! So you multiply all this by two to get the total area of both shaded areas:\[2(4-\pi)\]\[8-2\pi\]That's the answer in terms of pi.