Discovering Euler's Formula
You can use the various 3D shapes below to collect information about the faces, edges, and vertices for several prisms and pyramids. Use this digital resource to engage with "Investigating Relationships for 3D Shapes Labsheet".
For each of the following prisms, count the number of faces, edges, and vertices and record your information on your lab sheet. You can move the shapes around to help you investigate the shape. As you work, what patterns and relationships do you notice?
Prism A
Prism B
Prism C
Prism D
Try to generalize a pattern.
Look at your table of values. What patterns and relationships do you notice between the faces, vertices and edges for each prism?
Make a prediction: What number of faces, vertices and edges do you think a decagonal prism (10-sided base) will have?
Check your prediction by looking at the prism below (Prism E).
Prism E
Now let's investigate several pyramids. Do you think the same relationship will hold true for pyramids? Why or why not?
Pyramid A
Pyramid B
Prism C
What patterns and relationships are you noticing? Why might the patterns you are observing occur?
What would you predict would be the number of faces, vertices, and edges for an octagonal pyramid? (Look below to check your prediction).
Pyramid D
Extend your thinking!
Would your pattern that you found (also known as Euler's Formula) for pyramids and prisms work for 3D shapes that have a circular base (cone and cylinder)?
Why or why not? Justify your thinking.