1. Hook
If you were to slice the Great Pyramid of Giza with a geometric plane as shown, what shape would result?
Coach’s Commentary
I chose this example because it gives students a familiar object to envision.

2. Objective
In this lesson you will learn how to describe the cross sections of a right rectangular pyramid by slicing at different angles.

3. Let’s Review
Review geometric figures: right rectangular pyramid, plane.
Coach’s Commentary
Proper use of vocabulary is essential as students move to higher levels of mathematics.

4. A Common Mistake
A common mistake is interpreting a perspective drawing as though you were looking at it straight on.
In this case, the top face of this right rectangular prism appears to be a parallelogram. However, if we stand it up and view it straight on, we can see that it is a rectangle.
By keeping this in mind, we can begin to visualize more complex three-dimensional images.
Coach’s Commentary
Visualizing three-dimensional objects is difficult for some students. You may have to supplement this lesson with some physical models.

5. Vocabulary: Cross section
The two-dimensional shape that results from cutting a three-dimensional shape with a plane
Core Lesson
Imagine the surface of the water in a cylindrical drinking glass.
When we look at it from the side, the cross section appears to be an oval, but if we look at it straight on, we can see that the cross section of a cylinder is a circle.
We can imagine cross sections of right rectangular prisms in a similar way.
Coach’s Commentary
A familiar situation such as this can help students move to more complex images.

6. Core Lesson
Parallel planes – two planes that never intersect, like the floor and the ceiling of a room.

7. Core Lesson
Perpendicular planes – two planes that intersect to form a right (90 degree) angle, like the floor and one wall of a room.

8. Core Lesson
If we were to cut a right rectangular pyramid with a plane parallel to its base, what would be the shape of the resulting cross section?
Observe how the plane moves across the pyramid and try to imagine the resulting cross section.
Notice the blue shaded region – this is the cross section.
From this perspective, it looks like a parallelogram, but if we stand it up and look at it straight on, we can see that it is a rectangle.
Coach’s Commentary
This series of animations was designed to help students visualize the “cutting” process step by step.

9. Core Lesson
Now, let’s cut a right rectangular pyramid with a plane not parallel to its base.
What would be the shape of the resulting cross section this time?
Observe how the plane moves across the pyramid and try to imagine the resulting cross section.
Notice the blue shaded region – this is the cross section.
From the original perspective, the cross section appears to be a trapezoid; if we stand it up and look at it straight on, we can see that it is in fact a trapezoid.

10. Core Lesson
Now let’s consider a plane that is perpendicular to the base of the pyramid and contains its vertex. The vertex is the point where the triangular faces intersect.
What shape do you think this cross section will have?
Again we see the plane intersecting with the pyramid.
Can you visualize the cross section?
Now we can see the cross section in blue, and conclude that it is a triangle.

11. Review
In this lesson you have learned how to describe the cross sections of a right rectangular pyramid by slicing at different angles.

12. Guided Practice
Describe the shape of the cross section formed by the intersection of this plane and pyramid, if the plane does not contain the vertex of the pyramid.
Observe how the plane moves through the pyramid. Try to visualize the resulting cross section.
We can draw it in and then rotate it to look at it straight on.
A trapezoidal cross section is the result.

13. Extension Activities
Explain why any plane through the vertex of a pyramid will intersect the pyramid to form a triangular cross section.
Coach’s Commentary
Explaining why this happens will help students to solidify their understanding. It may be helpful to provide the student with a physical model.

14. Extension Activities
For a student who gets it and is ready to be challenged further: Describe the shape of the cross section formed by the intersection of a right rectangular pyramid and a plane that intersects the base and only two of the triangular faces of the pyramid.

16. Answer:
2. The plane would intersect all four triangular faces of the pyramid, but not be parallel to the base.