Why is the volume of a sphere divided by 3?

Answer 1

( The volume of a sphere is (4/3)(pi)r3 ).

The short answer: because of calculus.

The long answer:

This can be seen by using calculus to derive the volume of a sphere from the formula from it's surface area. To do this, we imagine that the sphere is full of infinity thin spheres inside it (all centered at the big sphere's center), and add up the surface areas of all the spheres inside.

The formula for the surface area of a sphere is 4(pi)r2. Let's call R the radius of the big sphere we want to find the volume of. To find the volume of this sphere, we add up the surface areas of all the spheres whose radii range from 0 to R. This gives the following formula (where r is the radius of each little sphere):

0R∫ 4(pi)r2dr

The 4 and pi can be factored out giving:

4(pi) (0R∫r2dr)

Integrating gives:

4(pi) [r3/3]0R

This is where the three comes from. Finishing the evaluation of the integral gives:

4(pi)(R3/3 - 03/3) = 4(pi)(R3/3)

Which can be rewritten as

(4/3)pi(R3)

which is the formula for the volume of a sphere.