Volume of Regular Cone and Truncated Cone (Frustrum)

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Volume of a Cone

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The small elemental volume dV has a height of dh and radius r, \[\frac{R}{H}=\frac{r}{h}\] Differential volume: \[dV=A(h)\,dh\] \[dV=\pi r^2\,dh\] \[dV=\pi \frac{R^2}{H^2} h^2 dh \] \[V=\int dV=\pi \frac{R^2}{H^2} \int_0^H h^2 dh = \pi \frac{R^2}{H^2} \left[\frac{h^3}{3}\right]_{0}^{H} \]

\[\boxed{V=\frac{1}{3}\pi R^2 H} \]

Volume of Truncated Cone

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The small elemental volume dV has a height of dh and radius r, \[\frac{R_1}{H+H_2}=\frac{R_2}{H_2}=\frac{r}{h+H_2}\] \[R_1H_2=R_2H+R_2H_2\] \[(R_1-R_2)H_2=R_2H\] \[H_2=\frac{R_2H}{R_1-R_2}\] To write changing radius r as a function of height h, \[\frac{R_2}{H_2}=\frac{r}{h+H_2}\] \[\frac{R_1-R_2}{H}=\frac{r-R_2}{\,h+\dfrac{R_2H}{R_1-R_2}\,}\] \[r=\frac{h(R_1-R_2)+R_2H}{H}\] \[r=R_2+\frac{(R_1-R_2)}{H}\,h\] Differential volume element Cross-sectional area: \[A(h)=\pi \,r^2\] Differential volume: \[dV=A(h)\,dh\] \[dV=\pi r^2\,dh\] Substituting for r: \[dV=\pi\left[R_2+\frac{(R_1-R_2)}{H}h\right]^2dh\] Expanding: \[dV=\pi\left[R_2^2+\frac{(R_1-R_2)^2}{H^2}h^2+\frac{2R_2(R_1-R_2)}{H}h\right]dh\] Total volume of the frustum \[V=\int_0^H\pi\left[R_2^2+\frac{(R_1-R_2)^2}{H^2}h^2+\frac{2R_2(R_1-R_2)}{H}h\right]dh\] Integrating: \[V=\pi\left[R_2^2h+\frac{(R_1-R_2)^2}{H^2}\frac{h^3}{3}+\frac{2R_2(R_1-R_2)}{H}\frac{h^2}{2}\right]_{0}^{H}\] \[V=\pi H\left[R_2^2+\frac{(R_1-R_2)^2}{3}+R_2(R_1-R_2)\right]\] Combining terms: \[V=\frac{\pi H}{3}\left[3R_2^2+(R_1-R_2)^2+3R_2(R_1-R_2)\right]\] Expanding: \[V=\frac{\pi H}{3}\left[R_1^2+R_1R_2+R_2^2\right]\] Final Result

\[\boxed{V=\frac{\pi H}{3}\left(R_1^2+R_1R_2+R_2^2\right)}\]