In the figure below let be the area of the shaded region and the area of the triangle . Compute

- ;
- ;
- .
- First, to calculate the area of the triangle we calculate the area of the polygon and subtract the area of the triangle . Since the polygon is two congruent triangles (since the line from to bisects the polygon into two equal sized triangles) we have
To calculate the area of the triangle we use the auxiliary point . The line from to is the height of the triangle and it has length . Therefore,

Hence,

- The area of the shaded region is the area of circular sector minus the area of the triangle (which we calculated in part (a) as ). The area of the sector is given by
Hence,

- For this we take the limit using our expressions for and obtained in parts (a) and (b) and use L’Hopital’s,
Then, we note that (from the trig identity ). Therefore,

We have the following diagram, with some additional markings from the diagram in Apostol,