Sphere - Volume of A Sphere

Volume of A Sphere

In 3 dimensions, the volume inside a sphere (that is, the volume of a ball) is derived to be

where r is the radius of the sphere and π is the constant pi. This formula was first derived by Archimedes, who showed that the volume of a sphere is 2/3 that of a circumscribed cylinder. (This assertion follows from Cavalieri's principle.) In modern mathematics, this formula can be derived using integral calculus, i.e. disk integration to sum the volumes of an infinite number of circular disks of infinitesimally small thickness stacked centered side by side along the x axis from x = 0 where the disk has radius r (i.e. y = r) to x = r where the disk has radius 0 (i.e. y = 0).

At any given x, the incremental volume (δV) is given by the product of the cross-sectional area of the disk at x and its thickness (δx):

The total volume is the summation of all incremental volumes:

In the limit as δx approaches zero this becomes:

At any given x, a right-angled triangle connects x, y and r to the origin, hence it follows from the Pythagorean theorem that:

Thus, substituting y with a function of x gives:

This can now be evaluated:

Therefore the volume of a sphere is:

Alternatively this formula is found using spherical coordinates, with volume element

In higher dimensions, the sphere (or hypersphere) is usually called an n-ball. General recursive formulas exist for the volume of an n-ball.

For most practical purposes, the volume of a sphere inscribed in a cube can be approximated as 52.4% of the volume of the cube, since . For example, since a cube with edge length 1 m has a volume of 1 m3, a sphere with diameter 1 m has a volume of about 0.524 m3.