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

A circle is tangent to the two smaller sides of a rectangle with dimensions a, 2a. What is the area of the surface enclosed by the rectangle but not by the circle?

The diameter of the circle spans the longer side of the rectangle and, therefore, the radius of the circle is:

equation

If the radius of the circle and the width of the rectangle are equal in length, then the shaded sector in the figure below has a central angle of 60°.

figure

The area needed can be computed by subtracting the areas of the two sectors OAB, ACD and of the two triangles DOA, BOC from the area of the rectangle.

The area of the triangle DOA is:

equation

The area of the sector AOB is:

equation

There are two congruent triangles and sectors that have to be subtracted:

equation

Simplifying:

equation