A Strange Solid

The base is a circle with radius r, now consider a sliver of the circle at distance x from the center of the circle (to the right or to the left, it doesn't matter) of width dx. The length of the sliver is $\sqrt{r^{2} - x^{2}}$ . This is also the base of the equilateral triangle. The volume of this triangle $= dV = \sqrt{3} \times (r^{2} - x^{2}) \times dx$ . Integrating from 0 to r, we get $V = \frac{2}{3} \times \sqrt{3} \times r^{3}$ . However this is only half the volume. The total volume $= \frac{4}{3} \times \sqrt{3}$ thus, $A+B+C = 10$