I need to find the area of the shaded area. The triangle is equilateral. So far, I have found the area of the triangle to be $\sqrt 3$, but I cannot figure out how to find the radius of the circle in order to find the area of the circle. Any advice would be appreciated.
[Math] How to find the area of the shaded area of this circle
geometry
Best Answer
Let's label Ahmed's drawing: Triangle $ABC$, lower left $A$, then counterclockwise $B$, and $C$ (top). Let the center of the circle be $M$. Extend $CM$ to intersect $AB$ in $D$. Note length $AD$ $=$ length $DB$ $=1$, $MD$ being the perpendicular bisector of $AB$. Triangle $ADM$ is a right angled triangle. Angle $MAD = 30°$.
$$\cos (30°) = \frac{1}{r}$$
$$r = \frac{1}{\cos (30°)}$$
Using $\cos (30°) = \frac{\sqrt{3}}{2}$ we get $r$.