Volume - Volume Formula Derivations - Cone

Cone

The cone is a type of pyramidal shape. The fundamental equation for pyramids, one-third times base times altitude, applies to cones as well.

However, using calculus, the volume of a cone is the integral of an infinite number of infinitesimally small circular slabs of thickness dx. The calculation for the volume of a cone of height h, whose base is centered at (0,0,0) with radius r, is as follows.

The radius of each circular slab is r if x = 0 and 0 if x = h, and varying linearly in between—that is,

The surface area of the circular slab is then

The volume of the cone can then be calculated as

and after extraction of the constants:

Integrating gives us

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