Lesson 13Polyhedra

Learning Goal

Let’s investigate polyhedra.

Learning Targets

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

  • I can explain the difference between prisms and pyramids.

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

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: Assembling Polyhedra

Problem 1

Your teacher will give you the net of a polyhedron. Cut out the net, and fold it along the edges to assemble a polyhedron. Tape or glue the flaps so that there are no unjoined edges.

Problem 2

How many vertices, edges, and faces are in your polyhedron?

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.

Two polyhedra showing the edge, face, and vertex of each. One is a rectangular prism and the other is a square triangular prism. Both have blue face.

A polyhedron always encloses a three-dimensional region.

The plural of polyhedron is polyhedra. Here are some drawings of polyhedra:

3 polyhedra

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 triangular prism, a pentagonal prism, and a rectangular prism.

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 rectangular pyramid, a hexagonal pyramid, a heptagonal pyramid, and a decagonal pyramid.

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

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.

A cube has 6 square faces, so its net is composed of six squares, as shown here.

A net can be cut out and folded to make a model of the polyhedron.

In a cube, every face shares its edges with 4 other squares. In a net of a cube, not all edges of the squares are joined with another edge. When the net is folded, however, each of these open edges will join another edge.

It takes practice to visualize the final polyhedron by just looking at a net.