# Mensuration-46

### From Homeworkwiki

**If h, c and V be the height, curved surface and volume of a cone, show that**

**3 Π V h ^{3} - c^{2} h^{2} + 9V^{2} = 0**

**Solution:** Let r be the radius and l the slant height of the cone.

Then c = curved surface = Π r l = Π r √(h^{2} + r^{2})

V = Volume = 1/3 x Π r^{2} h

Given expression = 3 Π V h^{3} - c^{2} h^{2} + 9V^{2}

= 3 Π (1/3 x Π r^{2} h) h^{3} - Π^{2} r^{2} (h^{2} + r^{2}) h^{2}

+ 9 (1/9 Π^{2} r^{4} h^{2})

= Π^{2} r^{2} h^{4} - Π^{2} r^{2} h^{4} -

Π^{2} r^{4} h^{2} + Π^{2} r^{4} h^{2} = 0 = R.H.S.