Here is a circle with a radius of 3.
It has 4 triangles cut out of it.
What is the remaining area of the shaded in region?
Note that the four triangles EGO, EIO, IHO, and HGO can be rotated, along with their shading, about the line segments EG, EI, IH, and HG respectively to transform FIG. 1 into FIG. 2. So the blue regions in the two figures have equal area. The area of the blue regions in FIG. 2, however, can be calculated easily. Each side of the square has the same length as the radius of the circle, so the shaded (blue) region has area equal to
\(\pi r^2-r^2\) \(=9\pi-9\) square units.
The side of the square = 3√2
So the legs of the small white triangle at the top = (1/2) of this = (3/2)√2
So....the area of one of the small white triangle = (1/2) [(3/2)√2]^2 = 9/4 units^2
So.....the area of the 4 small white triangles= 4 (9/4) = 9 units^2
And the area of the circle = pi (3)^2 = 9pi units^2
So......the shaded area =
[ 9pi - 9] units^2 =
9 [ pi - 1 ] units^2 ≈ 19.27 units^2