Define Original Volume

The volume of a sphere with radius $r$ is given by the formula $V = \frac{4}{3} \pi r^3$.

$V_{old} = \frac{4}{3} \pi r^3$

Define New Radius

The radius is halved, so the new radius $r'$ is $r' = \frac{r}{2}$.

$r' = \frac{r}{2}$

Calculate New Volume

Substitute $r'$ into the volume formula to find the new volume $V_{new}$.

$V_{new} = \frac{4}{3} \pi (\frac{r}{2})^3 = \frac{4}{3} \pi (\frac{r^3}{8})$

Compare Volumes

Simplify the expression to see the relationship between $V_{new}$ and $V_{old}$.

$V_{new} = \frac{1}{8} \times (\frac{4}{3} \pi r^3) = \frac{1}{8} V_{old}$