jmc

Question

A semicircle of diameter 1 sits at the top of a semicircle of diameter 2, as shown. The shaded area inside the smaller semicircle and outside the larger semicircle is called a

lune

. Determine the area of this lune. Express your answer in terms of

π

and in simplest radical form.

Accepted Answer

First note that the area of the region determined by the triangle topped by the semicircle of diameter 1 is

\frac{1}{2} \cdot \frac{3}{2} + \frac{1}{2} π (\frac{1}{2})^{2} = \frac{3}{4} + \frac{1}{8} π .

The area of the lune results from subtracting from this the area of the sector of the larger semicircle,

\frac{1}{6} π (1)^{2} = \frac{1}{6} π .

So the area of the lune is

\frac{3}{4} + \frac{1}{8} π - \frac{1}{6} π = \frac{3}{4} - \frac{1}{24} π .

Note that the answer does not depend on the position of the lune on the semicircle.