In the previous section, you investigated cross sections and slices of prisms and cylinders – 3-dimensional figures with 2 congruent, parallel bases. In this section, you will investigate cross sections and slices of pyramids and cones – 3-dimensional figures with only one base.

To begin with, let’s investigate pyramids. 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.

1. Click the radio button next to Pyramid.


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


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 pyramid?


6. Create a table like the one shown. 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 pyramid and the shape of the cross-section? (Reminder: a cross-section is parallel to the base of a pyramid). Why do you think that relationship is true?

7. Return to the triangular pyramid 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 pyramid 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 pyramid? Use a table like the one shown to summarize your findings.
   You may create the table in your notes.


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

Journal Entry:

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

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

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