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Question

An annulus is the region between two concentric circles. The concentric circles in the figure have radii b and c, with b>c. Let OX be a radius of the larger circle, let XZ be tangent to the smaller circle at Z, and let OY be the radius of the larger circle that contains Z. Let a=XZ, d=YZ, and e=XY. What is the area of the annulus? FIGURE

Accepted Answer

The area of the large circle is b^2, the area of the small one is c^2, hence the shaded area is (b^2-c^2). From the Pythagorean Theorem for the right triangle OXZ we have a^2 + c^2 = b^2, hence b^2-c^2=a^2 and thus the shaded area is a^2.