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Cross-Sections and Nets

Cross-sections are intersections of a plane with a solid, and nets are unfolded, flat representations of the sides of a 3-D shape.

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Cross-Sections and Nets

Let's Think About It

License: CC BY-NC 3.0

Michael and Amy bought Christmas gifts for their family.  The mall customer service center provided two types of boxes, but they're not sure on the final shape.  The boxes provided were in a flat stack for construction at home.  Michael and Amy need to figure out the shapes of the boxes when they are constructed so they know which gifts will fit in which boxes.  

In this concept, you will learn how to identify the 3-dimensional figure that a net represents.

Guidance

License: CC BY-NC 3.0
       
Credit: Quinn Dombrowski
Source: https://flic.kr/p/fiheCr
License: CC BY-NC 3.0
       
License: CC BY-NC 3.0

If you look at the solid figures above, you can see the parts of the figures that are toward you; however, you cannot see the parts that are on the backside and underneath or above.  If you want to see the entire figure, then you need to draw a representation of the figure that is two-dimensional, or planar.  You can think of this as unfolding or taking apart the solid so that you can see all of the parts.

Look back at the three figures. The first one is a rectangular prism. If you want to draw this as a two-dimensional figure, then you have to think about all of the parts of the rectangular prism.  There are four edges of the base of a rectangular prism, so you know that there are six faces.

Look at how this box has been unfolded. You can see the pattern that it makes.  If you picture it in your mind, you can see how you could fold up the box again. The edges form the lines and you fold it along the dotted lines to create a rectangular prism once again.

The second figure is a square pyramid. Let’s think about its parts and how it might look if it was unfolded.  The base is a square and there are four triangles on the sides. You can unfold it from the vertex down. Here is what it looks like.

You can see that if you fold on the dotted lines, the faces are rejoined, forming a vertex at the top of the figure.

The last figure is a cylinder. You know that the base and the top of a cylinder are circles. Think about the middle as a rectangle that is wound around the edges of the circles. 

Now let's look at a cone.

License: CC BY-NC 3.0

When looking at a cone from the front, you see a triangle. When you look from the bottom, you see a circle.  When looking at the cone from above, you see a circle with a point in the center.  The point represents the top of the cone.  The circle represents the base.

License: CC BY-NC 3.0

Imagine the triangle that you see from the front wrapping around the circular base, forming the vertex at the top.  The net of a cone looks like this:

License: CC BY-NC 3.0

Nets let you see the shape of the base, as well as the side or faces, of a solid figure all at once.

Guided Practice

Draw the net of a pentagonal prism.

License: CC BY-NC 3.0

First, draw the top and base of the pentagonal prism, remembering that the top and base of any prism is the same shape.

License: CC BY-NC 3.0
License: CC BY-NC 3.0

Next, draw the rectangular side faces of the pentagonal prism, remembering that there are five edges on the base (and top) and that this is the same as the number of faces on the side.

License: CC BY-NC 3.0

The answer is that the net of a pentagonal prism looks like this:

License: CC BY-NC 3.0

Examples

Example 1

What solid figure is represented by the following net?

License: CC BY-NC 3.0

First, observe that there is one six-sided figure - a hexagon - representing the base of the figure.

Next, observe that there are six triangles that make up the side faces of the figure.

Then, observe that the six triangles meet in the center which will form a vertex at the top when folded.

The answer is the solid figure represented by the net above is a hexagonal pyramid.

Example 2

Draw the net of a triangular prism.

License: CC BY-NC 3.0

First, draw the top and base of the triangular prism, remembering that the top and base of any prism is the same shape.

License: CC BY-NC 3.0
       
License: CC BY-NC 3.0

Next, draw the rectangular side faces of the triangular prism, remembering that there are three edges on the base (and top) and that this is the same as the number of faces on the side.

License: CC BY-NC 3.0

Then, draw each of the different sections as one picture, with the base and top triangles in the appropriate positions around the side rectangles.

The answer is that the net of a triangular prism looks like this:

License: CC BY-NC 3.0

 

Follow Up

License: CC BY-NC 3.0

Remember Michael and Amy's unknown boxes?

With regard to the first box:

First, all of the faces are rectangles.

Next, there are four rectangular side faces.

Then, there are two rectangular end faces.  When folded, these represent the base and top.

The answer for the first box is it is a rectangular prism.

With regard to the second box,

First, all of the faces are squares.

Next, there are four square side faces.

Then, there are two square end faces. When folded, these represent the base and the top.

The answer for the second box is it is a cube.

Video Review

Explore More

Identify each solid by its net.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. Draw an example of a new net for each of the figures in numbers 1–10.

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 10.3. 

Vocabulary

cross section

cross section

A cross section is the intersection of a three-dimensional solid with a plane.
Net

Net

A net is a diagram that shows a “flattened” view of a solid. In a net, each face and base is shown with all of its dimensions. A net can also serve as a pattern to build a three-dimensional solid.
Polyhedron

Polyhedron

A polyhedron is a solid with no curves surfaces or edges. All faces are polygons and all edges are line segments.
Solid Figure

Solid Figure

A solid figure is a three-dimensional figure with height, width and depth.
Volume

Volume

Volume is the amount of space inside the bounds of a three-dimensional object.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ Credit: Quinn Dombrowski; Source: https://flic.kr/p/fiheCr; License: CC BY-NC 3.0
  4. [4]^ License: CC BY-NC 3.0
  5. [5]^ License: CC BY-NC 3.0
  6. [6]^ License: CC BY-NC 3.0
  7. [7]^ License: CC BY-NC 3.0
  8. [8]^ License: CC BY-NC 3.0
  9. [9]^ License: CC BY-NC 3.0
  10. [10]^ License: CC BY-NC 3.0
  11. [11]^ License: CC BY-NC 3.0
  12. [12]^ License: CC BY-NC 3.0
  13. [13]^ License: CC BY-NC 3.0
  14. [14]^ License: CC BY-NC 3.0
  15. [15]^ License: CC BY-NC 3.0
  16. [16]^ License: CC BY-NC 3.0
  17. [17]^ License: CC BY-NC 3.0
  18. [18]^ License: CC BY-NC 3.0
  19. [19]^ License: CC BY-NC 3.0

Explore More

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