Proportional triangles in upside down cones

The height and radius of the truncated cone will be proportional to the original height and radius. Let $V, r, h$ be the volume, radius and height of the original cone, and $V^*, r^*, h^*$ be the volume, radius and height of the truncated cone. If $\alpha$ is the constant of proportionality, then $r^* = \alpha r$ and $h^* = \alpha h$ . Since for a cone, $V \propto r^2h$ , it follows that

$\begin{array}{rcccl} \dfrac{V^*}{V} &= \ \ \alpha^3 &= \ \ \dfrac{1}{2} \\ \\ \implies & \ \ \alpha &= \ \ \sqrt[3]{\dfrac{1}{2}} &\approx \ \ 0.79 &= \ \ \boxed{79\%} \end{array}$