Lesson 14Nets and Surface Area

Learning Goal

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

Learning Targets

  • I can match polyhedra to their nets and explain how I know.

  • When given a net of a prism or a pyramid, I can calculate its surface area.

Lesson Terms

  • base (of a prism or pyramid)
  • face
  • net
  • polyhedron
  • prism
  • pyramid
  • surface area

Warm Up: Matching Nets

Problem 1

Each of the following 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.

  1. net with a square in the center and four triangles extending from each side.
  2. net made of four attached rectangles with two squares extending from the sides of the second rectangle.
  3. triangle shaped net made of four small triangles
  4. net made of three attached rectangles with a triangle attached to either side of the middle rectangle.
  5. net made of 6 squares in a t shape
  1. cube
  2. rectangular prism
  3. square pyramid
  4. triangular pyramid
  5. triangular prism

Activity 1: Using Nets to Find Surface Area

Problem 1

Your teacher will give you the nets of three polyhedra to cut out and assemble.

Three nets on a grid, labeled A, B, and C. Net A is composed of two rectangles that are 5 units tall by 6 units wide, two that are 5 units high and one unit wide, and two that are one unit high and six units wide. Net B is a square with a side length of 4 units and is surrounded by triangles that are four units wide at the base and four units high. Net C is a square with a side length of 3, a rectangle 3 units wide and 5 units high, another rectangle that is 3 units wide and 4 units tall, and two triangles, one on either side, that are three units tall by four units across.

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

    A:

    B:

    C:

  2. Cut out your nets and use them to create three-dimensional shapes.

  3. Find the surface area of each polyhedron. Explain your reasoning clearly.

Are you ready for more?

Problem 1

For each of these nets, decide if it can be assembled into a rectangular prism.

Four possible nets labeled A--D.

Problem 2

For each of these nets, decide if it can be folded into a triangular prism.

Four possible nets labeled A--D.

Lesson 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 .