Frustums — Worked Examples

A conical frustum has larger radius $R=5.00\,\text{m}$, smaller radius $r=3.00\,\text{m}$, and perpendicular height $h=6.00\,\text{m}$. Determine its volume.

Use the frustum from Example 1. Determine its slant height and curved lateral area.

For the frustum in Examples 1–2, determine the total closed surface area.

A conical frustum has $R=6.00\,\text{m}$, $r=2.00\,\text{m}$, and height $h=8.00\,\text{m}$. Determine the height $x$ of the small cone removed from the apex and the original full-cone height.

Use the reconstructed geometry from Example 4 to compute the frustum volume by subtraction.

A square pyramidal frustum has lower side $10.0\,\text{m}$, upper side $6.00\,\text{m}$, and perpendicular height $9.00\,\text{m}$. Determine its volume.

Use the square frustum from Example 6. Determine its lateral area.

A conical frustum has $R=4.00\,\text{m}$, $r=2.00\,\text{m}$, and required capacity $V=176\pi\,\text{m}^3$. Determine the perpendicular height.

A hollow conical-frustum shell has outer radii $R_o=3.00\,\text{m}$ and $r_o=2.00\,\text{m}$, inner radii $R_i=2.70\,\text{m}$ and $r_i=1.70\,\text{m}$, and common perpendicular height $5.00\,\text{m}$. Determine the material volume.

For a conical frustum with $R=5.00\,\text{m}$, $r=3.00\,\text{m}$, and $h=6.00\,\text{m}$, verify its volume using the end areas and midpoint area.

A conical-frustum formula is evaluated with $R=r=4.00\,\text{m}$ and $h=7.00\,\text{m}$. Show that it reduces to the cylinder volume.

A frustum has volume $40.0\,\text{m}^3$ and total surface area $55.0\,\text{m}^2$. A similar frustum is built with every linear dimension multiplied by $k=1.80$. Determine the new volume and area.