How Much Is Yellow

Let r be half of the original radius and $A_T$ be the total area of yellow. Let $A_c$ be the area of quarter circle with radius $r$ . Let $A_1$ be half of the yellow part centered by point y. There are 3 $A_1$ here, 2 of them has their right angle connected to y and the other 1 is connected to x. So the total area of yellow is $A_T =A_c + 3A_1$ $A_1$ is the difference between area of $r \times r$ square and area of quarter circle with radius $r$ . So $A_1 = r^{2} - {A_c}$ Then:

$\begin{aligned} A_T &=A_c + 3A_1 \\ &= A_c + 3(r^{2} - A_c) \\ &= A_c + 3r^{2} - 3A_c \\ &= 3r^{2} - 2A_c \\ &= 3r^{2} - 2\left(\frac{\pi r^{2}}{4}\right) \\ &= 3r^{2} - \frac{\pi r^{2}}{2} \\ &= r^{2}(3 - {\pi \over 2}) \\ \end{aligned}$

At $r = {10 \over 2} = 5$ , $\begin{aligned} A_T &= (5)^{2}(3 - {\pi \over 2}) \\ &\approx 35.73009183013 \\ &\approx 35.7 \end{aligned}$

$Area\quad of\quad the\quad intersection\quad \\ of\quad two\quad half\quad circles\quad =\quad a,\\ \\ Smaller\quad half\quad circle\quad =\quad c\quad \\ \\ and\quad the\quad quarter\quad circle\quad =\quad b\\ \\ Now\quad area\quad of\quad the\quad blue\quad portion\quad \\ \qquad \quad \quad \quad =\quad b-(2c-a)+2a\\ \\ so\quad Area\quad of\quad the\quad yellow\quad portion\quad \\ \qquad \qquad =\quad b-(b-(2c-a)+2a)\\ \qquad \qquad =\quad 2c-3a\\ \\ Now\qquad \qquad \qquad \qquad \\ \qquad \quad \quad \quad c\quad =\quad \frac { 25\Pi }{ 2 } \quad =\quad 39.26\\ \\ for\quad finding\quad a\quad consider\quad the\quad quarter\quad arc\quad \\ of\quad the\quad smaller\quad circlewhich\quad includes\quad \\ the\quad intersection\quad area\quad a\quad \\ \\ \qquad \qquad a\quad =\quad 2(\frac { c }{ 2 } \quad -\quad \triangle XCM)\quad \\ \frac { c }{ 2 } \quad is\quad the\quad area\quad of\quad the\quad quarter\quad arc\quad \\ of\quad half\quad small\quad circle\quad and\quad M\quad is\quad the\quad \\ \\ contact\quad point\quad of\quad the\quad 2\quad cicles\quad at\quad perifery\\ \quad \quad \\ So\qquad \qquad \\ \qquad \qquad a\quad =\quad 2*(\frac { 25\Pi }{ 4 } -\frac { 25 }{ 2 } )\quad =\quad 14.27\quad \\ \\ Hence\quad the\quad acrea\quad of\quad the\quad yellow\quad portion\quad \\ \qquad =\quad 2c-3a\quad =\quad 2*39.26-3*14.27\quad =\quad 35.7$