Question

Question #9757e

Answer 1

I assume, the area `S=233pi` is given, not circumference.
Then
`V~~4742.12pi`

In terms of given circumference `C`,
`V=C/(6pi^2)`

Explanation:

The formula for volume of a sphere of a radius `R` is
`V=4/3piR^3`

The formula for area of a sphere of the same radius is
`S=4piR^2`

Given the surface area, we can find a radius from the last formula:
`R=sqrt(S/(4pi))= sqrt((233pi)/pi)=sqrt(233)`

Now we can fund volume:
`V=4/3piR^3=4/3pi(sqrt(233))^3~~4742.12pi`

If, instead of an area, circumference of the equator (the largest circle on a surface of a sphere) is given, the calculations are:
`C=2piR`
`R=C/(2pi)`
`V=4/3piR^3=4/3pi(C/(2pi))^3=4/3piC^3/(8pi^3)=C^3/(6pi^2)`