Overlapping Area

You have done a good job obtaining the answer. However many steps could be shortened.
$\alpha = \arcsin \left(\sqrt{ \dfrac{\frac{12 - {a^2} \times \pi }{12 \times \pi } } {\frac{1}{\pi }} }\right)\\ =\arcsin \left(\sqrt{~~~ \dfrac{12 - \frac4{\sqrt3} \times \pi }{12 } ~~~}\right)\\ =\arcsin \left(\sqrt{1 - \frac \pi{3\sqrt3}}\right)\\$
I missed since I took $\alpha ~in~ place~of~2\alpha.$ . My solution is as under.

LaTeX: $\begin{array}{l} S_T = \frac{\sqrt 3 }{4} \times {a^2} \Rightarrow a = \sqrt {\frac{4 \times S_T}{\sqrt 3 }} = \sqrt {\frac{4}{\sqrt 3 }} = 2 \times \sqrt {\frac{1}{\sqrt 3 }} \\ S_C = S = \pi \times {r^2} \Rightarrow r = \sqrt {\frac S\pi} = \sqrt {\frac{1}{\pi }} \\ \tan(30) = \frac{ | SM|}{\frac a2} = \frac{2 \times | SM|} a \Rightarrow | SM | = \frac{a \times \tan ( 30)} 2 = \frac{a \times \frac 1{\sqrt 3 }} 2 = \frac a{2 \times \sqrt 3 }\\ |CM| = \sqrt {a^2 - {\left( \frac a 2 \right)}^2} = \sqrt {a^2 - \frac{a^2}4} = \sqrt {\frac 3 4 \times a^2} = a \times \frac{\sqrt 3 } 2\\ | CS| = |CM| - |SM| = \frac{a \times \sqrt 3 } 2 - \frac a{2 \times \sqrt 3 } = \frac{3 \times a - a} {2 \times \sqrt 3} = \frac{2 \times a}{2 \times \sqrt 3 } = \frac a{\sqrt 3 }\\ |NS| = \sqrt {|CS|^2 - ( \frac a 2)^2} = \sqrt {\frac{a^2} 3 - \frac{a^2} 4} = \sqrt {\frac{a^2}{12}} = a \times \sqrt {\frac 1{12} } \\ | NR| = \sqrt {r^2 - |NS|^2} = \sqrt { \left( \sqrt { \frac {1}{ \pi} } \right)^2 - \left( a \times \sqrt { \frac 1 {12} } \right)^2 }= \sqrt {\frac 1 \pi - \frac{a^2}{12}} = \sqrt {\frac{12 - a^2 \times \pi }{12 \times \pi }} \\ \alpha = \arcsin \left(\sqrt{ \dfrac{\frac{12 - {a^2} \times \pi }{12 \times \pi } } {\frac{1}{\pi }} }\right) =\arcsin \left(\sqrt{~~~ \dfrac{12 - \frac4{\sqrt3} \times \pi }{12 } ~~~}\right)=\arcsin \left(\sqrt{1 - \dfrac \pi{\sqrt{27}} }\right)\\ S_{ARC} = \dfrac{1}{2} \times \left( \sqrt {\dfrac 1 \pi } \right)^2 \times \left( \dfrac{\pi \times 2 \times \alpha }{180} - \sin ( 2 \times \alpha)\right)\\ = \dfrac{1}{2 \times \pi } \times \left( \dfrac{\pi \times 2 \times \alpha }{180} - \sin ( 2 \times \alpha ) \right) = \dfrac{\pi \times \alpha }{180 \times \pi } - \dfrac{\sin ( 2 \times \alpha )}{2 \times \pi }\\ = \dfrac{\pi \times \alpha - 90 \times \sin ( 2 \times \alpha )}{180 \times \pi }\\ S = S_C - 3 \times S_{ARC} = 1 - 3 \times \dfrac {\pi \times \arcsin \left( \sqrt { 1 - \dfrac{\pi }{\sqrt {27} } } \right)- 90 \times \sin \left( 2 \times \arcsin \left( \sqrt { 1 - \frac{\pi }{\sqrt {27}} }\right) \right)}{180 \times \pi } \sim 0.817528694... \end{array}$
Above is the same program whose latex your editor has given. But your editor has inserted many many
unnecessary { } and \left \right, making it very difficult to follow. You may go for a better editor. Since I write
directly in latex I have no idea of any editor. One thing you can do even with your editor. After every \\ add three or more spaces and a return. In this way your lines even in latex will be separated. With best wishes.

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