What is the formula for the volume of a sphere?

What is the formula for the volume of a sphere?

• A. V=1/3Bh
• B. V=4/3πr³
• C. V=1/3πr²h
• D. A=πr²

Answer: (B)

Volume of a sphere comes from how its cross-sections look: at any distance from the center, the slice is a circle with area π(r^2 − x^2). Summing all those circular slices from one end of the sphere to the other gives V = ∫_{−r}^{r} π(r^2 − x^2) dx, which evaluates to (4/3) π r^3. This shows why the radius is the only size parameter and why the volume scales with the cube of the radius; doubling the radius makes the volume eight times larger. The other formulas correspond to different shapes: 1/3 times base area times height is the general cone/pyramid formula, and for a cone with a circular base that becomes V = (1/3) π r^2 h. The expression π r^2 is the area of a circle, not a volume.

Volume of a sphere comes from how its cross-sections look: at any distance from the center, the slice is a circle with area π(r^2 − x^2). Summing all those circular slices from one end of the sphere to the other gives V = ∫_{−r}^{r} π(r^2 − x^2) dx, which evaluates to (4/3) π r^3. This shows why the radius is the only size parameter and why the volume scales with the cube of the radius; doubling the radius makes the volume eight times larger.

The other formulas correspond to different shapes: 1/3 times base area times height is the general cone/pyramid formula, and for a cone with a circular base that becomes V = (1/3) π r^2 h. The expression π r^2 is the area of a circle, not a volume.