Introduction to Solid Mensuration — Worked Examples

Convert $2.50\,\text{m}^3$ to cubic centimeters.

A regular hexagon has side length $4.00\,\text{m}$. Determine its area for use as the base of a prism.

A proposed convex polyhedron is reported to have $10$ vertices, $18$ edges, and $10$ faces. Check whether these counts satisfy Euler's relation.

A storage-tank model is built at a linear scale of $1:10$. The model holds $5.00\,\text{L}$. Assuming perfect geometric similarity, determine the full-scale geometric capacity.

Two similar concrete blocks have corresponding surface areas $150\,\text{cm}^2$ and $600\,\text{cm}^2$. A corresponding height of the smaller block is $10.0\,\text{cm}$. Determine the matching height of the larger block.

A right prism and an oblique prism each have base area $7.50\,\text{m}^2$ and perpendicular height $4.00\,\text{m}$. At every corresponding level parallel to the bases, their sections have area $7.50\,\text{m}^2$. Determine and compare their volumes.

A solid has parallel end areas $A_1=20.0\,\text{m}^2$ and $A_2=44.0\,\text{m}^2$. Its midsection halfway between the ends has area $A_m=31.0\,\text{m}^2$, and the perpendicular distance between the end sections is $12.0\,\text{m}$. Assuming the prismatoidal formula applies, determine the volume.

A concrete component has geometric volume $3.20\,\text{m}^3$. If the assumed concrete density is $2400\,\text{kg/m}^3$, determine the mass represented by that geometric model.

A water-filled geometric volume is $12.0\,\text{m}^3$. Using water specific weight $\gamma=9.81\,\text{kN/m}^3$, determine the water weight.

A proposed volume formula for a family of solids is $V=\pi r h$. Determine whether it can represent a volume when both $r$ and $h$ are lengths.