# Lesson 11

Polyhedra and Nets

### Lesson Narrative

In this lesson, students learn about polyhedra and their nets. They also study prisms and pyramids as types of polyhedra with certain defining features. Then, students practice visualizing the polyhedra that could be assembled from given nets and use nets to find the surface area of polyhedra. When exmaining nets, students must reason abstractly and quantitatively (MP2) to mentally manipulate the figure into three dimensions to determine the type of polyhedron.

Polyhedra can be thought of as the three-dimensional analog of polygons.

Here are some important aspects of polygons:

• They are made out of line segments called edges.
• Edges meet at a vertex.
• Edges only meet at vertices.
• Polygons always enclose a two-dimensional region.

Here is an analogous way to characterize polyhedra:

• They are made out of filled-in polygons called faces.
• Faces meet at an edge.
• Faces only meet at edges.
• Polyhedra always enclose a three-dimensional region.

Students do not need to memorize a formal definition of a polyhedron, but help them make sense of nets and surface area.

### Learning Goals

Teacher Facing

• Compare and contrast (orally and in writing) features of prisms and pyramids.
• Comprehend and use the words “face”, “edge”, “vertex”, and “base” to describe polyhedra (in spoken and written language).
• Use a net with gridlines to calculate the surface area of a prism or pyramid and explain (in writing) the solution method.

### Student Facing

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

### Required Preparation

• Assemble collections of geometric figures that each contains at least 2 familiar polyhedra, 2 unfamiliar polyhedra, and 2 non-polyhedra. Prepare one collection for each group of 3–4 students.  If pre-made polyhedra are unavailable, assemble some from the nets in the blackline master for the warm-up.
• Print and pre-cut the nets and polygons in the blackline master for Prisms and Pyramids. Prepare 1 set per group of 3–4 students, along with tape to join the polygons into a net.
• Make copies of the nets in the blackline master for the activity Using Nets to Find Surface Area. Prepare one of the 3 nets (A, B, and C) and some glue or tape for each group of 3 students.

### Student Facing

• 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.

Building Towards

### 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.

• 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.

• 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.