圓錐台體積
\[ \displaystyle \begin{aligned} &= \text{原始大圓錐體積} - \text{被切掉的小圓錐體積} \\[6pt] &= \frac{1}{3} \times \left( \text{大圓面積} \times \text{大圓錐高} - \text{小圓面積} \times \text{小圓錐高} \right) \\[6pt] &= \frac{1}{12} \; \pi \left[ D^2 H' - d^2 \left( H' - H \right) \right] \\[6pt] &= \frac{1}{12} \; \pi \left[ \left( D^2 - d^2 \right) H' + d^2 H \right] \\[6pt] \\ &\text{由於相似三角形} \\[12pt] &\to \frac{\text{大圓錐高} \, H'}{\text{大圓直徑} \, D} = \frac{\text{大圓錐高} \, H' - \text{圓錐台高} \, H}{\text{小圓直徑} \, d} \\[12pt] &\to \text{大圓錐高} \, H' = \frac{D}{D - d} \, H \\[6pt] \\ &\text{將大圓錐高}\, H' \,\text{代回原式, 可得圓錐台體積} \\[6pt] &= \frac{1}{12} \; \pi \left[ \left( D^2 - d^2 \right) \frac{D}{D-d} \, H + d^2 H \right] \\[6pt] &= \frac{1}{12} \; \pi \left[ \left( D + d \right) D H + d^2 H \right] \\[6pt] &= \frac{1}{12} \; \pi H \left( D^2 + Dd + d^2 \right) \\[6pt] \end{aligned} \] 