Here is an old geometry/measurement chestnut for you.

[polymath257]

Draw a quarter circle and label the center point C and the points on either end of its arc, A and B.

Let the length of the quarter circle's radius = 1 unit 

Now draw the circle with diameter AB, and label another point on this circle D such that point D is not inside the original quarter circle CAB.

Find the area of circle ABD outside of quarter circle CAB.  Place answer inside hide tags and justify your result.

An old problem of Hipparchus.

---

The area of the half-circle ABD is the same as the area of the quarter circle. Subtract the area outside of the triangle and the quarter circle from both.
The answer is the same as the area of the triangle OAB where O is the center of the quarter circle. In other words, 1/2.

---