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Cross-Sections and Nets

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What if you wanted to expand your thinking of geometric shapes beyond the flat two-dimensional ones to three dimensional (3D) ones? In this chapter we are going to expand to 3D. Copy the equilateral triangle to the right onto a piece of paper and cut it out. Fold on the dotted lines. What shape do these four equilateral triangles make? If we place two of these equilateral triangles next to each other (like in the far right) what shape do these 8 equilateral triangles make? After completing this Concept, you'll be able to answer questions like these.

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CK-12 Foundation: Chapter11CrossSectionsandNetsA

Guidance

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.

Example A

What kind of figure does this net create?

The net creates a rectangular prism.

Example B

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.

Example C

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

The cross-section is a circle.

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter1CrossSectionsandNetsB

Concept Problem Revisited

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

Vocabulary

A cross-section is the intersection of a plane with a solid. A net is an unfolded, flat representation of the sides of a three-dimensional shape.

Guided Practice

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

1.

2.

3.

Answers:

1. Square

2. Rhombus

3. Hexagon

Practice

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.

  1. A cube has 11 unique nets. Draw 5 different nets of a cube.
  2. 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?
  3. Describe a method to construct a net for a regular octahedron.
  4. Can you tell what a polyhedron looks like from looking at one cross section?

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