Properties of a Sphere

The Sphere

• Volume

Volume

Similar to the derivation of the area of a circle, the volume of a sphere can be obtained through calculus. We treat the volume as the continuous sum of the areas of infinitesimally thin discs, the radii of which vary with z, with the radius of the sphere, R, being the point of maximum z.

`r=Rcostheta` `z=Rsintheta`

We first express the area of each infinitesimal disc as a function of its radius,

`A=pir^2`

We next need to find r in terms of z,

`arcsin(z/R)=theta`

Hence,

`r=Rcostheta=Rcos(arcsin(z/R))`

Recalling cos(arcsin(x)) and sin(arccos(x)) are related via the unit triangle, ie.,

`cos(arcsin(x))=sin(arccos(x))=sqrt(1-x^2)`

Then we have,

`r=Rsqrt(1-z^2/R^2)`

Hence the area of each infinitesimal disc can be expressed in terms of z,

`A=pir^2=pi R^2(1-z^2/R^2)`

Integrating this expression with respect to z, therefore yields the volume, which we note is only half of a sphere,

`V/2=int_0^(z=R) pi R^2(1-z^2/R^2) dz`

We expand the integral,

`V/2=pi R^2(int_0^(z=R) dz - int_0^(z=R) z^2/R^2 dz)= piR^2( [z]_0^R - 1/3[z^3]_0^R/R^2)`

...and insert the limits,

`V/2=piR^2( R - 1/3R)=pi( R^3 - 1/3R^3)=(2pi)/3R^3`

Hence,

`V=(4pi)/3R^3`