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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

On their second day at the mall, Candice and Trevor were taken to the back store room where there were tons of boxes. Only these boxes weren’t put together, they were all flat.

“This is where we keep boxes for wrapping,” Mrs. Scott told them.

“That is a lot of boxes,” Trevor said looking at all of the piles of boxes.

“Yes and they all have gotten mixed up. We need you to sort them. Put all of the square boxes in one pile and all of the rectangular boxes in another pile. Then we will be able to get to the box that we need when the time comes,” Mrs. Scott instructed.

“We can do that,” Candice said.

Trevor did not look as certain.

“Alright. I’ll come back and check on you in a little while,” Mrs. Scott said walking away.

“What do you mean, we can do that?” Trevor asked Candice after Mrs. Scott had left.

“It’s easy. The square ones look like the nets of cubes-you know from math class. The other boxes look like rectangular prisms.”

Trevor is puzzled. He can’t remember what a net for a cube looks like.

Do you know? What is the difference between a net for a cube and a net for a rectangular prism?

This Concept is all about nets and two-dimensional representations of three-dimensional figures. By the end of the Concept, you will know how to identify the net of cube from the net of a rectangular prism.

### Guidance

Think about the solid figures that you learned about in the Identify and Describe Faces, Edges and Vertices of Solid Figures Concept. Solid figures are three-dimensional figures that have height, width and length. Here are some of the solid figures that you learned about in the last Concept.

If you look at these figures, you can see that there are some parts that you can see and some parts that you can’t see. We can assume that the other parts are there, but because this is a solid figure we can only see parts of each figure.

What if we wanted to see the whole figure?

If we want to see the entire figure, then we need to draw a representation of the figure that is two-dimensional.

You can think of this as unfolding or taking apart the solid so that you can see all of the parts of the solid.

Look back at the three figures that we just looked at. The first one is a rectangular prism. If we wanted to draw this as a two-dimensional figure, then we would have to think about all of the parts of the rectangular prism.

There are four sides to the base of a rectangle, so we know that there are six faces. But how do the faces look? Let’s look at this drawing of a rectangular prism.

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

Let’s look at the second figure. It is a square pyramid. If we were to draw a square pyramid, let’s think about its parts and how it might look if we unfolded it.

The base is a square and there are four triangles on the sides. We can unfold it from the vertex down. Here is what it looks like.

You can see that the sides could be rejoined at the vertex and we would have the square pyramid once again.

What about the last figure? It is a cylinder. We know that the two bases of a cylinder are circles. What about the middle? Think about that middle space as a rectangle that is wound up around the edges of the circle. If you think about it this way, you can see what it would look like taken apart.

Two-dimensional representations of solids like this one is called a net. A net is a visual way of drawing a solid figure so that all of the parts of the figure can be seen and measured.

Nets, as we know, allow us to see the top, bottom, and sides of a solid figure all at once. We can also get a sense of a solid just by drawing its side, front, and top views. These are like pieces of a net, without having to draw the whole net.

As with nets, we’ll need to use our imagination to draw the side, top, and front view. Imagine you can hold the figure and can look down on it from above, or look at it from the side. This may be hard to imagine at first. If so, find some real solid objects, such as a soup can and cereal box.

What is the top view of this cylinder?

If you think about what you would see if you look down on this can, you would see the circle that is the base.

What about if we looked at it from the side or front?

Then you would see a rectangle of sorts. It would be long and represent the length of the can.

Look at this figure. It is a cereal box, but we can also think of it given its geometric properties. It is a rectangular prism. We know that a rectangular prism is made up of rectangles, but would we see a rectangle from each view? Yes. We would, but the position of the rectangle would change depending on the view.

Here is a top view.

Here is a side view.

Here is a front view and a back view.

You can imagine that because of the dimensions of the rectangle, that you would be able to see any of the different parts of the figure and draw them too.

Cones can be tricky because of what you would see. If you look at the side, you would see a triangle. If you look from the bottom up, you would see a circle.

What about from the top down?

The point would represent the top of the cone. The circle represents the bottom part.

We can draw a view of any three-dimensional figure as long as you take the time to visual it in your mind first. If you have trouble with this, try to use a real object to help you with the visualization and the drawing.

As we have seen, we can classify solid figures by drawing nets of them. The nets let us see what shape the base is, how many sides the figure has, and whether or not it has any parallel faces. In the last section, we practiced drawing different views of three-dimensional figures. Now we can use those views as we work to identify different solids. This is how we can identify and/or classify solid figures by using their side and top views.

For instance, we know from drawing pyramids that their sides are triangles. If we look at it from the side, then, all we can see is one triangular face. Pyramids always have triangular sides, so we would know just from this one side view that the figure must be a pyramid. What about this figure?

The square is a tricky one to work with. Any prism could have a square as one of its views. We could also have a square pyramid with a bottom up view, although this would not be as common.

Depending on the view, we can either be general or specific in our identification of the solid. You can see that the square is a more general view to work with.

What solid figure is represented below?

We have been given a side view and a top view.

What shape are the side faces of this solid figure? They are rectangles. What shape is the top face of the figure? It is a circle. Now let’s think about what we solid figures we know. Many solid figures can have rectangular sides (any prism), but not very many have a circular top face. Cylinders, as we know, have a circular top face and a circular base. This could be a cylinder.

But what about the side? The side of a cylinder is actually curved. Remember, when we “unroll” it to make a net, the side looks like a rectangle. Imagine you could hold up a can of soup. It would look like a rectangle from the side!

These two views must be showing a cylinder.

When given different views, you will need to think about each part of the different solids. Keep in mind that knowing that there are prisms, cylinders, cones and pyramids can help you in this task.

We can use cubes called “unit cubes” to help us draw different views of a solid figure. This is a way to “build” a solid figure. This is where we are going to begin to see some of the dimensions of different solid figures come into practice.

This is the building block that we are going to use for our drawings. Here is how this is going to work.

Draw the top view of rectangular prism. The dimensions are units.

To draw this figure, we are going to use cubes. We know that the top view is units. We take the cubes and build the view of the rectangular prism.

This is what the top view of the rectangular prism would look like.

What if we were to know all of the dimensions of a rectangular prism? Could we build it then?

Yes, we could. To know all of the dimensions, we would have the length, the width and the height. Then we would use cubes to “build the different views” of the figure.

Whew! We have really looked at solids inside and out! Now we can take them apart and put them together again, all by understanding the shapes of their faces and how the faces fit together. In the next few lessons you will see how useful it is to understand solids.

Now it's time for you to try a few on your own.

#### Example A

Draw a net of a pentagonal prism.

#### Example B

Draw a net of a cube.

#### Example C

Draw a net of a cylinder.

Here is the original problem once again. Reread it and then see how Candice explains the difference between the net of a cube and the net of a rectangular prism.

On their second day at the mall, Candice and Trevor were taken to the back store room where there were tons of boxes. Only these boxes weren’t put together, they were all flat.

“This is where we keep boxes for wrapping,” Mrs. Scott told them.

“That is a lot of boxes,” Trevor said looking at all of the piles of boxes.

“Yes and they all have gotten mixed up. We need you to sort them. Put all of the square boxes in one pile and all of the rectangular boxes in another pile. Then we will be able to get to the box that we need when the time comes,” Mrs. Scott instructed.

“We can do that,” Candice said.

Trevor did not look as certain.

“Alright. I’ll come back and check on you in a little while,” Mrs. Scott said walking away.

“What do you mean, we can do that?” Trevor asked Candice after Mrs. Scott had left.

“It’s easy. The square ones look like the nets of cubes-you know from math class. The other boxes look like rectangular prisms.”

Trevor is puzzled. He can’t remember what a net for a cube looks like.

Candice looks at Trevor. She picks up one of the boxes.

“Okay, see this one, it is a square box or you could think of it as the net of a cube. Look at the center of the box. All four sides are the same length. When we fold up the sides around this bottom, you can see that all of the sides match. They are the same length and so they will form a cube.”

Trevor smiles.

“Okay, now I get it. The rectangular box has a rectangle at the center. The sides will fold up around it.”

“Yes,” Candice adds. “We can see that there is a length and a width for the rectangular bottom and top. Also, notice that the sides match these lengths. Therefore, when we fold up the sides, we will have a box that is a rectangular prism.”

With this explanation complete, the students got right to work sorting all of the boxes.

### Vocabulary

Solid Figure
a three-dimensional figure with height, width and length.
Net
a visual way to represent a three-dimensional figure in a two-dimensional.

### Guided Practice

Here is one for you to try on your own.

Name this figure.

This is the net of a rectangular prism.

### Practice

Directions: 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.

### Vocabulary Language: English

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.