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This problem trains for: SAT-I, AMC-8, AMC-10, Math Kangaroo 9-10.
A circle of radius 2 units is tangent interior to a circle of radius 6 units. One of the radii that delimit a circular sector with central angle 144° in the larger circle, passes through the point of contact of the two tangent circles. What is the area of the set of points that are inside the sector but outside of the small circle?
The point of contact of two tangent circles is on the same line as the centers of the circles.
As a result, the sector has an intersection with only half of the small circle.
Does the semicircle lie entirely inside the sector or not?
To answer this question let us figure out the central angle that is circumscribed to the semicircle:
The central angle is such that:
Therefore, the semicircle is completely contained inside the sector.
The area in question is obtained by subtracting the area of the small semicircle from the area of the sector.
The area of the sector is:
The area of the semicircle is:
The desired area is: