Question

A cone, a hemisphere and a cylinder stand on equal bases and have the same height. Show that their volumes are in the ratio 1:2:3.

Answer 1

If all of have the same base.

Then: radius is same in all the solids.

Vol of Cone = 1/3πr²h

Vol of cylinder = πr²h

Vol of hemisphere = 2/3πr³

Their ratio is: = 1/3πr²h : πr²h : 2/3πr³

= 1/3 : 1 : 2/3r

Thus,

The ratio is 1/3:1:2/3

= 1:3:2

= Cone:Cylinder:Hemisphere

Answer 2

The cone , the hemisphere and the cylinder has the same radius and the same height .

Let their radii be r .

Let their height be h .

So the volumes of the figure are :

Cone = 1/3 π r² h

Cylinder = π r² h

Hemisphere = 2/3 π r³

We know that they have the same height and base .

Hence we can say that r = h .

This value should be put in the above formulas :

Cone = 1 / 3 π r² × r

⇒ 1 / 3 π r³

Cylinder = π r² × r

⇒ π r³

Hemisphere

⇒ 2 / 3 π r³

Ratio will be :

1 / 3 π r³ : 2 / 3 π r³ : π r ³

Multiply every ratio by 3 π r³

⇒ 1 : 2 : 3

Hence proved !