Semicircles on the Square

First, find the area of the total square

$A_s = 20^2 = 400$

Next, subdivide each corner of the square into a smaller square of side 5 (One square at each point where two semicircles meet, with diagonal to the nearest corner). We can then find the area of one orange section within this sub-square, by finding the area of the sub-square and subtracting the area of half of one of the semi-circles:

$A_{\square, orange} = A_{\square} - \frac{\pi 5^2}{4}$

Using this information we can then easily find the area of the orange segment of the square because it is composed of 8 quarter-circles, and 8 of these orange sub-square sections. Thus the area of the yellow and red section is:

$A_{total} = A_s - A_{orange} = 400 - (8\cdot A_{\frac{1}{4}-circle} + 8*A_{\square, orange})$

$A_{total} = 400 - 8\cdot (\frac{\pi\cdot 5^2}{4} + (5^2 - \frac{\pi\cdot 5^2}{4})) = 400 - 8\cdot 5^2\cdot (1) = 400 -200 = 200$

shortcut: (hahaha)

$20^{2} - 8*(5^{2}) = 200$

The equation of the circle is x^2+y^2=25. Rearrange, y=sqrt(25-x^2). Next, integrate the function to find the area under the curve from 0 to 5. You get 19.635. Take this is minus the area of the triangle of 25/2 from this and you have 7.135 unit square. Each red region is therefore 2* 7.135=14.27. Next, take the area of the square ( 20 20) of 400 minus the are of the semi circles (( 25 p/2) 8) of 314.159. You get 85.841 unit square. Since we remove the red regions earlier twice for each, we need to add them back twice. 14.27 4*2=114.16 unit sq. Lastly, add them both together 85.841+114.16=200