Polyhedra and Prisms — Worked Examples

A cube has $8$ vertices, $12$ edges, and $6$ faces. Verify Euler's formula.

A convex polyhedron has $V=10$ vertices and $E=15$ edges. Determine the number of faces.

A closed rectangular prism measures $4.00\,\text{m}\times3.00\,\text{m}\times5.00\,\text{m}$. Determine its volume and total surface area.

A right-triangular prism has base legs $3.00\,\text{m}$ and $4.00\,\text{m}$ and prism height $10.0\,\text{m}$. Determine its volume and total surface area.

An oblique prism has base area $18.0\,\text{m}^2$, perpendicular height $7.00\,\text{m}$, and lateral edges $8.50\,\text{m}$ long. Determine its volume.

A prism has constant base area $B=12.5\,\text{m}^2$ and must enclose $100\,\text{m}^3$. Determine the required perpendicular height.

A concrete cube has side length $5.00\,\text{m}$. Determine its volume and total surface area.

A cube has total surface area $294\,\text{cm}^2$. Determine its side length and volume.

A cube side is doubled. Determine the factors by which its total surface area and volume change.

A triangular base has area $B=24.0\,\text{m}^2$. A plane cuts a right prism so the three heights measured at the base vertices are $4.00$, $6.00$, and $8.00\,\text{m}$. Determine the volume when the cutting-plane height varies linearly over the triangular base.

A rectangular concrete block is $1.20\,\text{m}\times0.800\,\text{m}\times0.500\,\text{m}$. If concrete density is $2400\,\text{kg/m}^3$, estimate its mass.

A rectangular prism has base dimensions $6.00\,\text{m}\times2.00\,\text{m}$ and height $5.00\,\text{m}$. A right-triangular prism has base legs $6.00\,\text{m}$ and $4.00\,\text{m}$ and the same height. Compare their volumes.