Prism nets, surface area, and volume with worked examples

A prism can look complicated until we unfold it. Its net shows which faces to add for surface area, and its base and height tell us how much space is inside.

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Find the two matching bases

A prism has two congruent, parallel polygonal bases. Its other faces are parallelograms. In a right prism, those side faces are rectangles, and its height is the perpendicular distance between the bases. A triangular prism has triangular bases, and a rectangular prism has rectangular bases.

Check which faces meet when the net folds

A net lays all the faces flat. For a right triangular prism, look for two matching triangles and three rectangles. The triangles must close opposite ends, and the widths of the three rectangles around them must match the three sides of each triangle.

Which net folds into a pyramid? Count the shapes and see how their edges can meet.

A and B both fold into triangular prisms. Their triangles close opposite ends even though they attach in different places. C has four triangular faces and folds into a triangular pyramid. D folds into a rectangular prism.

Add all six faces to find surface area

For a rectangular prism with length $l$, width $w$, and height $h$, the three pairs of faces have areas $lw$, $lh$, and $wh$. A box measuring $6$ by $2$ by $3$ centimeters has the faces shown here.

The labels give the side lengths of each face in centimeters.

$\begin{array}{l}S=2lw+2lh+2wh\\ S=2(12+18+6)=72{\mathrm{cm}}^2\end{array}$

For any right prism with base area $B$, base perimeter $P$, and height $h$, the rectangles around the outside have total area $Ph$. Adding both bases gives $S=2B+Ph$.

Multiply one base area by the height to find volume

The same base shape extends through the whole prism. Its volume is $V=Bh$, where $h$ is measured perpendicular to the bases.

For our rectangular prism, use the $6$ by $2$ face as the base, with area $12{\mathrm{cm}}^2$. The distance to the opposite base is $3\mathrm{cm}$.

$V=\left(12\right)\left(3\right)=36{\mathrm{cm}}^3$

The units are cubic because we multiplied an area by a length. For a triangular prism, find the area of one triangle first and then multiply it by the perpendicular distance between the triangles.

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