# 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:

Here are pictures that do not represent polyhedra:

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

Here are some polyhedra called pyramids.

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

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.

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

### 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 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.

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. 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. 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.

• 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.

• 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.

• 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.

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.