The purple area is the sum of the circle sector and the triangle:
[tex]A = A_s+A_t[/tex]
Let's compute them one at the time:
The sector is identified by an angle of 260°, because it is the remainder of a 100° angle.
We can build this simple proportion
[tex]\text{total area}\div\text{total angle}=\text{sector area}\div\text{sector angle}[/tex]
The area of the circle is [tex]\pi r^2[/tex], so we have
[tex]\pi 8.35^2\div 360= A_s\div 260[/tex]
Solving for the sector area, we have
[tex]A_s = \dfrac{\pi\cdot8.35^2\cdot260}{360} = \dfrac{8.35^2\cdot 13\pi}{18}[/tex]
The triangle is an isosceles triangle, because two of the sides are radii. This means that the height is also a bisector, so we can cut the triangle in two 90-50-40 triangles.
Using the law of sines, we can deduce that the height is
[tex]8.35\sin(40)[/tex]
And half the base is
[tex]8.35\sin(50)[/tex]
So, the area of the triangle is
[tex]A_t = 8.35^2\sin(40)\sin(50)[/tex]
So, the purple area is
[tex]A = A_s+A_t =\\\dfrac{8.35^2\cdot 13\pi}{18}+8.35^2\sin(40)\sin(50) = 8.35^2\left(\dfrac{13}{18}\pi+\sin(40)\sin(50)\right)\approx 192.5[/tex]
