What if you wanted to expand your thinking of geometric shapes beyond the flat two-dimensional ones to three dimensional (3D) ones?

Try this experiment to gee how nets relate to 3D figures: Sketch or print the equilateral triangle in the image below onto a piece of paper and cut it out. Fold on the dotted lines. What shape do these four attached equilateral triangles make? If you place two of these equilateral triangles next to each other, as in the second image, and fold them on the dotted lines, what 3D figure would you make?

### Cross-Sections and Nets

While our world is three dimensional, we are used to modeling and thinking about three dimensional objects on paper (in two dimensions). There are a few common ways to help think about three dimensions in two dimensions. One way to “view” a three-dimensional figure in a two-dimensional plane, like this text, is to use cross-sections. A **cross-section** is the intersection of a plane with a solid. Another way to represent a three-dimensional figure in a two dimensional plane is to use a net. A **net** is an unfolded, flat representation of the sides of a three-dimensional shape.

#### Identifying Figures Created by Nets

What kind of figure does this net create?

The net creates a rectangular prism, like a shallow rectangular box.

#### Drawing Nets

Draw a net of the right triangular prism below.

This net will have two triangles and three rectangles. The rectangles are all different sizes and the two triangles are congruent.

Notice that there could be a couple different interpretations of this, or any, net. For example, this net could have the triangles anywhere along the top or bottom of the three rectangles. Most prisms have multiple nets.

#### Describing Cross-Sections

Describe the cross section formed by the intersection of the plane and the solid.

The cross-section is a circle.

#### Earlier Problem Revisited

The net of the first shape is a regular tetrahedron and the second is the net of a regular octahedron.

### Examples

Describe the shape formed by the intersection of the plane and the regular octahedron.

#### Example 1

Square

#### Example 2

Rhombus

#### Example 3

Hexagon

### Review

Describe the cross section formed by the intersection of the plane and the solid.

Draw the net for the following solids.

Determine what shape is formed by the following nets.

- A cube has 11 unique nets. Draw 5 different nets of a cube.
- Use construction tools to construct a large equilateral triangle. Construct the three midsegments of the triangle. Cut out the equilateral triangle and fold along the midsegments. What net have you constructed?
- Describe a method to construct a net for a regular octahedron.
- Can you tell what a polyhedron looks like from looking at one cross section?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 11.2.