Four circles, each with a radius of 1 unit, are tangent to each other so that one circle lies fully within each quadrant. Each circle is tangent to exactly two other circles. On top of this configuration, a fifth circle (also with a radius of 1 unit) has a center point at the origin. See the screenshot provided. Find the area of the zone that lies fully within the center circle, and not within any other circle (the "star" shape). I spotted this pattern on one of my coworker’s shirts, and I figured it would be a fun challenge to work out the area of this portion of the pattern. However, it’s proving more difficult than I first imagined. I have a sense that the solution may involve squares and/or right triangles, but beyond that I have no idea.
Since $r=1$, the area of each circle $\pi r^2=\pi$, and the area of a square circumscribing each circle is $2^2=4$. So each square exceeds its circle by $4-\pi$, and has one-fourth of this excess in the region within the four circles.
Hence the area of the star is$$4 \left(\frac{4-\pi}{4}\right)=4-\pi\approx.858$$