Lesson 14

Nets and Surface Area

Let’s use nets to find the surface area of polyhedra.

14.1: Matching Nets

Each of the nets can be assembled into a polyhedron. Match each net with its corresponding polyhedron, and name the polyhedron. Be prepared to explain how you know the net and polyhedron go together.

Five nets of polyhedra labeled 1--5.
Five polyhedra labeled A--E.

 

14.2: Using Nets to Find Surface Area

  1. Name the polyhedron that each net would form when assembled.

    Three nets on a grid, labeled A, B, and C.
  2. Your teacher will give you the nets of three polyhedra. Cut out the nets and assemble the three-dimensional shapes.
  3. Find the surface area of each polyhedron. Explain your reasoning clearly.


  1. For each net, decide if it can be assembled into a rectangular prism.

    Four possible nets labeled A--D.

  2. For each net, decide if it can be folded into a triangular prism.

    Four possible nets labeled A--D.

Summary

A net of a pyramid has one polygon that is the base. The rest of the polygons are triangles. A pentagonal pyramid and its net are shown here.

The net for this pentagonal pyramid is a pentagon surrounded by triangles on each side.

A net of a prism has two copies of the polygon that is the base. The rest of the polygons are rectangles. A pentagonal prism and its net are shown here.

The net for this pentagonal prism is a pentagon surrounded by rectangles on each side with an additional pentagon attached to the opposite side of one of the rectangles.

In a rectangular prism, there are three pairs of parallel and identical rectangles. Any pair of these identical rectangles can be the bases.

Three images of a rectangular prism. Each image has one set of opposing sides of the polyhedron shaded and labeled “base."

Because a net shows all the faces of a polyhedron, we can use it to find its surface area. For instance, the net of a rectangular prism shows three pairs of rectangles: 4 units by 2 units, 3 units by 2 units, and 4 units by 3 units.

A polyhedron made up of six rectangles. Two rectangles are 8 square units in area, 2 are 6 square units, and 2 are 12 square units.

The surface area of the rectangular prism is 52 square units because \(8+8+6+6+12+12=52\).

Glossary Entries

  • base (of a prism or pyramid)

    The word base can also refer to a face of a polyhedron.

    A prism has two identical bases that are parallel. A pyramid has one base.

    A prism or pyramid is named for the shape of its base.

    Two figures, a pentagonal prism and a hexagonal pyramid.
  • face

    Each flat side of a polyhedron is called a face. For example, a cube has 6 faces, and they are all squares.

  • net

    A net is a two-dimensional figure that can be folded to make a polyhedron.

    Here is a net for a cube.

    Six squares arranged with 4 in a row, 1 above the second square in the row, and one below the second square in the row.
  • polyhedron

    A polyhedron is a closed, three-dimensional shape with flat sides. When we have more than one polyhedron, we call them polyhedra.

    Here are some drawings of polyhedra.

    3 polyhedra, from left to right shapes resemble a house, drum, and star.
  • prism

    A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.

    Here are some drawings of prisms.

    A triangular prism, a pentagonal prism, and a rectangular prism.
  • pyramid

    A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.

    Here are some drawings of pyramids.

    a rectangular pyramid, a hexagonal pyramid, a heptagonal pyramid
  • surface area

    The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron, without any gaps or overlaps.

    For example, if the faces of a cube each have an area of 9 cm2, then the surface area of the cube is \(6 \boldcdot 9\), or 54 cm2.