Which expression gives the volume of a sphere with radius r?

• A. Surface Area of the sphere
• B. Volume = base × height
• C. 4 π r^2
• D. 4/3 π r^3

Answer: (D)

When we measure how much space a sphere encloses, the radius controls the size, and volume grows with the cube of the radius. That’s why the volume formula has r cubed and a constant factor that comes from assembling all the cross-sectional disks.

If you imagine slicing the sphere into very thin circular disks perpendicular to a diameter, each disk at distance x from the center has radius sqrt(r^2 − x^2) and area π(r^2 − x^2). Summing up (integrating) all these disk areas from −r to r gives the total volume:

V = ∫_{−r}^{r} π(r^2 − x^2) dx = 4/3 π r^3.

So the expression for volume is 4/3 π r^3. The r^2 form would correspond to surface area, not volume, and the base × height idea isn’t the right rule for a sphere.