Problem:
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.
Answer Choices:
A. 61​π−43​​
B. 43​​−121​π
C. 43​​−241​π
D. 43​​+241​π
E. 43​​+121​π Solution:
First note that the area of the region determined by the triangle topped by the semicircle of diameter 1 is
21​⋅23​​+21​π(21​)2=43​​+81​π
The area of the lune results from subtracting from this the area of the sector of the larger semicircle,
61​π(1)2=61​π
So the area of the lune is
43​​+81​π−61​π=43​​−241​π​
Note that the answer does not depend on the position of the lune on the semicircle.
