Lesson 11

Polyhedra and Nets

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

11.1: What are Polyhedra?

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?

11.2: Prisms and Pyramids

  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?
  2. Which of these nets can be folded into Pyramid P? Select all that apply.
    Three nets. 

  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).



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

11.3: 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 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 prism and pyramid with labels indicating an edge, vertex, and face on each.

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 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.