# Mensuration-56

### From Homeworkwiki

**A cylinder is circumscribed along a hemisphere and a cone is inscribed in the cylinder.**

**The vertex of the cone is in the center of the base of the cylinder. Prove that**

**vol. of the cylinder / 3 = vol. of the hemisphere / 2 + vol. of the cone / 1**

**Solution:** Let r be the radius of the hemisphere. Then heights of the cone and cylinder are both r.

1/3 (vol. of the cylinder) = 1/3 (Π r^{2} r) = 1/3 Π r^{3}

1/2 (vol of hemisphere) = 1/2 (2/3 Π r^{3}) = 1/3 Π r^{3}

vol of the cone = 1/3 Π r^{2} r = 1/3 Π r^{3}

Therefore, vol of cylinder/3 = vol of hemisphere/2 = vol of cone/1