# 9.3: Cross-Sections and Nets

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Cross-sections are intersections of a plane with a solid. Nets are unfolded, flat representations of the sides of a 3-D shape that allow you to find the sum of the area of the faces by showing the edges and faces in 2-D.

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 on a piece of paper or a computer screen) is to use cross-sections. Another way to “view” a three-dimensional figure in a two-dimensional plane is to use a net.

Cross-Section: The intersection of a plane with a solid.

The cross-section of the peach plane and the tetrahedron is a triangle. Figure $$\PageIndex{1}$$

Net: An unfolded, flat representation of the sides of a three-dimensional shape. Figure $$\PageIndex{2}$$

It is good to be able to visualize cross sections and nets as the three dimensional objects they represent.

What if you were given a three-dimensional figure like a pyramid and you wanted to know what that figure would look like in two dimensions? What would a flat slice or an unfolded flat representation of that solid look like?

Example $$\PageIndex{1}$$

Describe the cross section formed by the intersection of the plane and the solid. Figure $$\PageIndex{3}$$

Solution

Circle

Example $$\PageIndex{2}$$

Determine what shape is formed by the following net. Figure $$\PageIndex{4}$$

Solution

Square-based pyramid

Example $$\PageIndex{3}$$

What is the shape formed by the intersection of the plane and the regular octahedron?

1. Figure $$\PageIndex{5}$$
2. Figure $$\PageIndex{6}$$
3. Figure $$\PageIndex{7}\ Solution 1. Square 2. Rhombus 3. Hexagon Example \(\PageIndex{4}$$

What kind of figure does this net create? Figure $$\PageIndex{8}$$

Solution

The net creates a rectangular prism. Figure $$\PageIndex{9}$$

Example $$\PageIndex{5}$$

Draw a net of the right triangular prism below. Figure $$\PageIndex{10}$$

Solution

The net will have two triangles and three rectangles. The rectangles are different sizes and the two triangles are the same. Figure $$\PageIndex{11}$$

There are several different nets of any polyhedron. For example, this net could have the triangles anywhere along the top or bottom of the three rectangles.

## Review

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

1. Figure $$\PageIndex{12}$$
2. Figure $$\PageIndex{13}$$

Draw the net for the following solids.

1. Figure $$\PageIndex{14}$$
2. Figure $$\PageIndex{15}$$
3. Figure $$\PageIndex{16}$$
4. Figure $$\PageIndex{17}$$
5. Figure $$\PageIndex{18}$$
6. Figure $$\PageIndex{19}$$

Determine what shape is formed by the following nets.

1. Figure $$\PageIndex{20}$$
2. Figure $$\PageIndex{21}$$
3. Figure $$\PageIndex{22}$$