Truncated Cone or Frustum Area and Volume

Truncated Cone Area and Volume

A truncated cone is the result of cutting an entire cone by a plane parallel to the base and removing the part containing the apex.

The height $h$ is the distance $O_{1}O_{2}$ between the centers of the bases.

Area of the truncated cone or frustum

To calculate the area of the truncated cone, we will consider the two cones:

-The entire cone with Base $O_1$ and apex $Q$

-The removed cone with Base $O_2$ and apex $Q$

-The difference will help calculate the lateral area and later the volume of the truncated cone.

The 2 slants:

$s_1=\sqrt{(h+x)^{2}+r_{1}^{2}}$

$s_2=\sqrt{x^{2}+r_{2}^{2}}$

TheLateral Areas:

$A_1=\pi r_{1}\sqrt{(h+x)^{2}+r_{1}^{2}}$

For the removed cone:

$A_2=\pi r_{2}\sqrt{x^{2}+r_{2}^{2}}$

The Lateral Area:

(1) $\begin{equation*}\begin{split}L.A&.=A_{1}-A_{2}\\&=\pi r_{1}\sqrt{(h+x)^{2}+r_{1}^{2}}-\pi r_{2}\sqrt{x^{2}+r_{2}^{2}}\end{split}\end{equation*}$

-Let’s find $x$

From the similar triangles above:

$\frac{h+x}{r_{1}}=\frac{x}{r_{2}}$

This yields:

$(h+x)r_{2}=r_{1}x$

$r_{2}h+r_{2}x=r_{1}x$

$x(r_{1}-r_{2})=r_{2}h$

$x=\frac{r_{2}h}{r_{1}-r_{2}}$

Back to the Lateral Area:

(2) $\begin{equation*}\begin{split}L.A.&=\pi r_{1}\sqrt{(h+\frac{r_{2}h}{r_{1}-r_{2}})^{2}+r_{1}^{2}}-\pi r_{2}\sqrt{\frac{r_{2}^{2}h^{2}}{(r_{1}-r_{2})^{2}}+r_{2}^{2}}\\&=\frac{\pi r_{1}}{r_{1}-r_{2}}\sqrt{(h(r_{1}-r_{2})+r_{2}h)^{2}+ r_{1}^{2}(r_{1}-r_{2})^{2}}-\frac{\pi r_{2}}{r_{1}-r_{2}}\sqrt{r_{2}^{2}h^{2}+r_{2}^{2}(r_{1}-r_{2})^{2}}\\&=\frac{\pi r_{1}}{r_{1}-r_{2}}\sqrt{(hr_{1}-hr_{2}+r_{2}h)^{2}+r_{1}^{2}(r_{1}-r_{2})^{2}}-\frac{\pi r_{2}}{r_{1}-r_{2}}\sqrt{r_{2}^{2}(h^{2}+(r_{1}-r_{2})^2)}\\&=\frac{\pi r_{1}}{r_{1}-r_{2}}\sqrt{h^{2}r_{1}^{2}+r_{1}^{2}(r_{1}-r_{2})^{2}}-\frac{\pi r_{2}}{r_{1}-r_{2}}\sqrt{h^{2}r_{2}^{2}+r_{2}^{2}(r_{1}-r_{2})^{2}}\\&=\frac{\pi r_{1}r_{1}}{r_{1}-r_{2}}\sqrt{h^{2}+(r_{1}-r_{2})^{2}}-\frac{\pi r_{2}r_{2}}{r_{1}-r_{2}}\sqrt{h^{2}+(r_{1}-r_{2})^{2}}\\&=\pi (r_{1}+r_{2})\sqrt{h^{2}+(r_{1}-r_{2})^{2}}\end{split}\end{equation*}$