Volumes of Solids

Volume measures the three-dimensional space an object occupies. For a container, its internal volume gives its storage capacity.

Imagine a plant manager calculating the total storage capacity of a chemical vessel featuring a cylindrical top and a cone-shaped bottom.

First, he analyzes the top cylinder, with 2-meter radius and a 6-meter height. The bottom of this section is a circle with an area of pi times radius squared.

The cylinder can be visualized as identical circular disks stacked vertically. Multiplying this base area by the 6-meter height gives the cylinder’s volume of approximately 75.4 cubic meters.

Next, he examines the 3-meter-tall conical base with the same 2-meter radius. Unlike the cylinder, the cone’s circular slices gradually shrink as they narrow toward the bottom drain.

Modeling this section as a right circular cone, it has one-third the volume of a matching cylinder, giving the conical base a volume of about 12.6 cubic meters.

Finally, to find the container's total capacity, he adds the volume of both sections together resulting in a combined storage volume of about 88.0 cubic meters.