Lesson 11Polyhedra and Nets

Learning Goal

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

Learning Targets

  • I can describe the features of a polyhedron using mathematical vocabulary.

  • I understand the relationship between a polyhedron and its net.

  • 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: What are Polyhedra?

Problem 1

Here are pictures that represent polyhedra:

5 polyhedra

Here are pictures that do not represent polyhedra:

a sphere, a cylinder, a strip with 3 twists joined end-to-end, and an open-top box.

  1. Your teacher will give you some figures or objects. Sort them into polyhedra and non-polyhedra.

  2. What features helped you distinguish the polyhedra from the other figures?

Activity 1: Prisms and Pyramids

Problem 1

Here are some polyhedra called prisms.

Six prisms, labeled A, B, C, D, E, and F.

Here are some polyhedra called pyramids.

Four polyhedral labeled P, Q, R, and S. Each figure has a base and a number of sides which share a single vertex.

  1. Look at the prisms. What are their characteristics or features?

  2. Look at the pyramids. What are their characteristics or features?

Problem 2

Which of the following nets can be folded into Pyramid ? Select all that apply.

  1. four small triangles arranged horizontally to create a parallelogram,
  2. four small triangles arranged to make a larger triangle.
  3. four small triangles which all meet at their vertices.

Problem 3

Your teacher will give your group a set of polygons and assign a polyhedron.

  1. Decide which polygons are needed to compose your assigned polyhedron. List the polygons and how many of each are needed.

  2. Arrange the cut-outs into a net that, if taped and folded, can be assembled into the polyhedron. Sketch the net. If possible, find more than one way to arrange the polygons (show a different net for the same polyhedron).

Are you ready for more?

Problem 1

What is the smallest number of faces a polyhedron can possibly have? Explain how you know.

Activity 2: 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 polyhedron is a three-dimensional figure composed of faces. Each face is a filled-in polygon and meets only one other face along a complete edge. The ends of the edges meet at points that are called vertices.

A rectangular prism showing parts of the face - edge and vertex. A triangular prism showing a face with vertex and edge noted.

A prism is a type of polyhedron with two identical faces that are parallel to each other and that are called bases. The bases are connected by a set of rectangles (or sometimes parallelograms). A prism is named for the shape of its bases. For example, if the base is a pentagon, then it is called a “pentagonal prism.”

A net is a two-dimensional representation of a polyhedron. It is composed of polygons that form the faces of a polyhedron. 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.

A pyramid is a type of polyhedron that has one special face called the base. All of the other faces are triangles that all meet at a single vertex. A pyramid is named for the shape of its base. For example, if the base is a pentagon, then it is called a “pentagonal pyramid.”

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.

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 net of a rectangular prism on a grid in different colors. The green is 8 square units, the blue is 12, and the orange is 6.

The surface area of the rectangular prism is 52 square units because .