What is the formula for the volume of a sphere?

• A. V = 4πr^3
• B. V = 4/3πr^3
• C. V = 2πr^3
• D. V = πr^3

Answer: (B)

The volume of a sphere grows with the cube of the radius, so you’ll see a r^3 term in the formula. A clear way to derive it is by slicing the sphere into thin horizontal disks. At a height y from the center, the slice is a circle with radius sqrt(r^2 − y^2), so its area is π(r^2 − y^2). To get the total volume, sum (integrate) these disk areas as y goes from −r to r: ∫_{-r}^{r} π(r^2 − y^2) dy.

Carrying out the integral gives π[r^2y − y^3/3] from −r to r, which simplifies to π[(r^3 − r^3/3) − (−r^3 + r^3/3)] = π[(2/3)r^3 − (−2/3)r^3] = (4/3)πr^3. This is the volume formula for a sphere.

As an alternative viewpoint, using spherical coordinates yields the same result, confirming the constant 4/3 and the r^3 dependence.