Geometric Measurement and Dimension: Informal Argument for Volume of a Sphere (CCSS.G-GMD.2)

Formulas: $V_{\text{cyl}}=\pi r^2 h$, $V_{\text{cone}}=\tfrac{1}{3}\pi r^2 h$, so the cone is one-third of the cylinder. By Cavalieri's principle, the cone's cross-sections scale so that its "average" cross-sectional area is $\tfrac{1}{3}$ of the cylinder's. The $\tfrac{1}{2}$ distractor confuses the cone's factor with a common but incorrect guess.

Formulas: $V_{\text{sphere}}=\tfrac{4}{3}\pi r^3$ and $V_{\text{cyl}}=\pi r^2(2r)=2\pi r^3$, so $\tfrac{V_{\text{sphere}}}{V_{\text{cyl}}}=\tfrac{4/3}{2}=\tfrac{2}{3}$. By Cavalieri's principle, slicing shows the sphere's cross-sections pair with those of a cone to match the cylinder, yielding the $\tfrac{2}{3}$ ratio. The "equal" and "greater" choices contradict these area-by-slices comparisons.

Sphere volume scales with $r^3$: $V=\tfrac{4}{3}\pi r^3$. Doubling $r$ multiplies volume by $2^3=8$. By Cavalieri's principle, all cross-sectional areas scale with $r^2$ and the overall depth with $r$, giving a cubic scaling. The $\times 4$ distractor mistakes area scaling for volume scaling.

Cylinder volume $V=\pi r^2 h$. Tripling $r$ multiplies cross-sectional area by $3^2=9$, and doubling $h$ multiplies by $2$, for a total factor $9\cdot 2=18$. By Cavalieri's principle, volume equals area of cross-sections times height, so area and height scalings multiply.

Formulas: $V_{\text{hemi}}=\tfrac{1}{2}\cdot \tfrac{4}{3}\pi r^3=\tfrac{2}{3}\pi r^3$ and $V_{\text{cone}}=\tfrac{1}{3}\pi r^2(r)=\tfrac{1}{3}\pi r^3$. Thus the hemisphere is twice the cone. By Cavalieri's principle, the hemisphere's slices average to double the cone's slices at matching heights.

Using formulas: $V_{\text{cone}}=\tfrac{1}{3}\pi r^2 h$ and $V_{\text{cyl}}=\pi r^2 h$, so the cone is one-third of the cylinder. The common mistake is $\tfrac{1}{2}$, not $\tfrac{1}{3}$. Cavalieri's principle supports the $\tfrac{1}{3}$ factor for pyramids/cones.

Formulas: $V_{\text{sphere}}=\tfrac{4}{3}\pi r^3$ and $V_{\text{cyl}}=\pi r^2(2r)=2\pi r^3$, so $\tfrac{V_{\text{sphere}}}{V_{\text{cyl}}}=\tfrac{4/3}{2}=\tfrac{2}{3}$. By Cavalieri, at each height the cross-sectional area of the sphere equals that of the cylinder minus a cone, giving the same $\tfrac{2}{3}$ result; $\tfrac{1}{2}$ is a common distractor.

For a sphere, $V=\tfrac{4}{3}\pi r^3$. Doubling $r$ multiplies volume by $2^3=8$. Confusing with surface area scaling leads to the incorrect factor of 4.

$V_{\text{cyl}}=\pi r^2(2r)=2\pi r^3$ and $V_{\text{cone}}=\tfrac{1}{3}\pi r^2(2r)=\tfrac{2}{3}\pi r^3$. So $V_{\text{cyl}}-V_{\text{cone}}=\tfrac{4}{3}\pi r^3$, which equals $V_{\text{sphere}}$. By Cavalieri, the sphere and the cylinder-minus-cone have equal cross-sectional areas at each height, so their volumes are equal.

For a cylinder, $V=\pi r^2 h$. Tripling $r$ multiplies volume by $3^2=9$ and tripling $h$ multiplies it by $3$, for a total factor of $9\times3=27$. Choosing 9 or 3 misses one of the dimensions.