# 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.

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by nneehhaa66

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by nneehhaa66

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So the vol. of the cone =

Vol. of hemisphere=

Vol.of the cylinder: [/tex]

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The formulas for the volumes of the solids are :

Cone:

V = 1/3 π R² H: R = radius of the base and H = the height of the cone

Base = π R²

Hemisphere:

V = 2/3 π R³ , R = radius

Base = π R²

Cylinder :

V = π R² H , R = radius and H is the height

Base = π R²

All of them have same base => Same Radius. They have same height too. The height of a hemisphere is same as the radius. Hence, R = H.

So ratio of their volumes = 1/3 π R² H : 2/3 π R³ : π R² H

= H : 2 R : 3 H

= 1 : 2 : 3

Cone:

V = 1/3 π R² H: R = radius of the base and H = the height of the cone

Base = π R²

Hemisphere:

V = 2/3 π R³ , R = radius

Base = π R²

Cylinder :

V = π R² H , R = radius and H is the height

Base = π R²

All of them have same base => Same Radius. They have same height too. The height of a hemisphere is same as the radius. Hence, R = H.

So ratio of their volumes = 1/3 π R² H : 2/3 π R³ : π R² H

= H : 2 R : 3 H

= 1 : 2 : 3