What is the area of the shaded part of the circle?
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In the diagram below, line segment $AT$ is a diameter of the circle with center $O$. What is the area of the shaded part of the circle?
$AT= 16$.
Half of the circles area is equal to $100.48$, on the other half of the circle there is a triangle spitting it up, The triangle is a right triangle with interior angle measures of $30, 60$, and $90$. There is one side known and that side is the diameter which is equal to $16$, it is the hypotenuse of the triangle.
$\endgroup$ 21 Answer
$\begingroup$Sides of a 30-60-90 triangle of hypotenuse $16$ are $8$ and $8 \sqrt{3}$. The area of that triangle is then $(1/2) (8)(8 \sqrt{3}) = 32 \sqrt{3}$. The shaded area is then the area of the circle minus the area of that triangle, or $64 \pi - 32 \sqrt{3}$.
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