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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10, SAT-II.

The inscribed angle ∠AMB=30° intercepts a circle of radius √2 at points A and B. What is the area of the shaded region?

figure

The shaded region is a circle segment that is subtended by a central angle of 60°. Denoting the center of the circle with O, the triangle OAB must be equilateral.

figure

To find the area of the shaded region we must subtract the area of the equilateral triangle OAB from the area of the circular sector OAB.

The sector is the sixth part of a circle, therefore its area is:

equation

The side length of the triangle and the height are in the ratio:

equation

Therefore, the height:

equation

equation

equation

and the area:

equation

equation