In this resource, you will investigate the intersection of a plane with different 3-dimensional figures. Depending on the angle of intersection and the type of 3-dimensional figure, there are certain possible shapes that the intersection could be.

A cross-section of a 3-dimensional figure is the shape that results when the plane is parallel to the base of the figure. A slice of a 3-dimensional figure is the shape that results when the plane is not parallel to the base of the figure. It could be perpendicular, or it could be an acute or obtuse angle with respect to the base.

In this section, you will use an interactive sketch to investigate patterns in cross sections and slices of prisms and cylinders.

To begin with, let’s investigate prisms. Click on the link, Interactive Cross Section Flyer. When the sketch opens, scroll down to the control bars and boxes beneath the 3-dimensional figure and the graph and follow the instructions.

1. Click the radio button next to Prism.


2. Click on the Lateral faces slider so that the prism becomes a triangular prism.


3. Click on the Rotate slice around Y slider until the slice, represented by the red portion of the figure, is parallel to the base.


4. Click on the Move slice slider. What do you observe as the slice moves up and down the figure? You can also click the Animate button to automatically move the slice.


5. What shape is the slice? How does this shape compare with the base of the prism?


6. Create a table like the one shown.

Prism	Shape of Cross Section
Triangular
Rectangular
Pentagonal
Hexagonal
Heptagonal
Octagonal

You may create the table in your notes. Increase the number of lateral faces, one at a time, using the Lateral faces slider. Record the shape of the cross-section in the table.

Pause and Reflect - Journal Entry

What relationship do you notice between the shape of the base of a prism and the shape of the cross-section? (Reminder: a cross-section is parallel to the base of a prism). Why do you think that relationship is true?

7. Return to the triangular prism using the Lateral faces slider.


8. Change the angle between the plane of intersection and the base by clicking on the Rotate slice around Y slider.


9. Move the slice (remember – it's not parallel to the base anymore, so it's technically not a cross-section) up and down the prism using the Move slice slider. Change the angle by clicking on the Rotate slice around Y slider. What other shapes can be generated by slicing a triangular prism? Use a table like the one shown to summarize your findings. You may create the table in your notes.

Shape of Slice	Intersection with Base and Lateral Faces

10. Repeat steps 8 and 9 for other prisms. What additional shapes can be generated by slicing other prisms?

Journal Entry

What patterns and relationships do you observe between the possible shapes of slices of prisms when the angle of intersection is not parallel to the base of the prism?

What is the minimum number of sides in the polygon created by slicing a prism at any angle? What is the maximum number of sides?

What relationships do you notice between the total number of faces in the prism and the number of sides in the polygons created by slicing the prism at angles that are not parallel to the base? Why do you think this is the case?