Why Volume of the Cone is one-third of the Cylinder

As we know, Both cylinder and cone are made up of circular plates arranged one above the other. Here the width of each circular plate inside both shapes is dx (very small).

In the case of a cylinder, the radius (r) of all the plates is the same from bottom to top but in the case of the cone, the radius of the plates decreases continuously from bottom to top and ends at zero at the top.

So, the Volume of each plate (Very Small)= Base Area × Height

dv=𝜋r².dx …………(i)

The Volume of the Cylinder= Sum of the Volumes of all the plates

After integrating both sides (Limit: 0 to h)

The volume of the Cylinder(v)= 𝝅r².h ……….. (ii)

In the case of Cone, △ABM~△AON (Similar Triangles)

So, AB/AO=BM/ON

Note: Here AB is the distance of the arbitrary plate from the Apex of the Cone. k is the radius of the plate (The radius of all the plates are different)

In the case of the bottom plate, x=h

Here x is associated with dx (The width of each plate)

x/h=k/r

k=rx/h ………(iii)

The Volume of the Cone= Sum of the Volumes of all the plates

After integrating both sides (Limit: 0 to h)

The volume of the Cone= 1/3.𝜋r²h

So, we can say the volume of the cone is one-third of the volume of a Cylinder of the same height and base radius.

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