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# Cross-Sections and Nets

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# Cross Sections of Solids

A pentagonal prism is sliced with a plane parallel to its base. The area of the cross section is 15 square inches. If the height of the pentagonal prism is 20 inches, what is the volume of the prism?

#### Guidance

A cross section is the shape you see when you make one slice through a solid. A solid can have many different cross sections depending on where you make the slice. Consider the following hexagonal pyramid.

Cross sections perpendicular to the base will be triangles. Below, you can see a plane cutting through the pyramid, part of the pyramid removed, and the cross section.

You could also take a slice parallel to the base. Cross sections parallel to the base will be hexagons.

It is also possible to take cross sections using planes that are neither parallel nor perpendicular to the base. These can be much more difficult to visualize. Physical models or dynamic geometry software are extremely helpful. For example, the same hexagonal pyramid has been sliced at a slant below. The cross section is a pentagon.

Cross sections are one way of representing three dimensional objects in two dimensions.

Example A

Using planes perpendicular or parallel to the base, what cross sections can you construct from a rectangular prism?

Solution: Considering only planes that are perpendicular or parallel to the base, all cross sections are rectangles. One example of this is shown below.

Example B

A square pyramid is sliced by a plane parallel to its base. Describe the cross section. How does moving the plane so that it is closer or further from the base change the cross section?

Solution: The cross section will always be a square. The closer the plane is to the base, the bigger the square.

Example C

Describe how to create a triangular cross section of a square pyramid.

Solution: Slice the pyramid with a plane that is perpendicular to the base of the pyramid.

Concept Problem Revisited

A pentagonal prism is sliced with a plane parallel to the base. This means that the cross section is a pentagon that is congruent to the base. The area of the cross section is 15 square inches. This means that the area of the base is 15 square inches. Since the height of the prism is 20 inches, the volume of the prism is $15 \cdot 20=300 \ in^3$ .

#### Vocabulary

A cross section is the shape you see when you make one slice through a solid.

#### Guided Practice

1. Describe the cross sections of a cylinder that are perpendicular or parallel to the base of the cylinder.

2. Could the cross section of a cylinder be an ellipse (oval)?

3. A cylinder is sliced parallel to its base. The area of the cross section is $36 \pi \ in^2$ . What is the radius of the cylinder?

1. Circles and rectangles.

2. Yes. If the slice is slanted as shown below:

3. The cross section is a circle with area  $36 \pi \ in^2$ . The radius of the circle is 6 inches, so the radius of the cylinder is 6 inches.

#### Practice

1. Describe the cross sections of a sphere.

2. Describe the cross sections of a pentagonal prism that are parallel or perpendicular to the base.

3. Describe the cross sections of a pentagonal pyramid that are parallel or perpendicular to the base.

4. Could the cross section of a cube be a triangle? Explain.

5. A cross section of a pyramid is taken parallel to its base. What are the connections between the cross section and the base?

6. A cross section of a pyramid is taken perpendicular to its base. What shape is the cross section? Does the shape of the base matter?

7. For a prism, information about what type of cross section can help you to determine the volume?

8. A cylinder with radius 4 inches is sliced parallel to its base. What is the area of the cross section?

9. The volume of a cylinder is $360 \pi \ in^2$ . The cylinder is sliced parallel to its base. If the height of the cylinder is 10 inches, what is the area of the cross section?

Use the solid below for 10-12.

10. A plane slices the solid parallel to its base. Describe the cross section. How does the cross section change as the plane moves further from the base?

11. A plane slices the solid perpendicular to its base through the center of the base. Describe the cross section.

12. Find the area of the cross section from #11.

Use the solid below for 13-15.

13. A plane slices the solid parallel to its base. Describe the cross section. How does the cross section change as the plane moves further from the base?

14. A plane slices the solid perpendicular to its base through the center of the base. Describe the cross section.

15. Find the area of the cross section from #14.