This activity center is used to illustrate the difference between the slant height and the actual perpendicular height (of the given square pyramid) by the use of the distance formula.
Before the Activity
I use this activity center after the students have finished a unit on the distance formula. I use it to illustrate the importance of the distance formula in analyzing coordinate geometry
During the Activity
First I have the activity center configured for the students to plot the vertices of the square pyramid (in front of the Louve in Paris) one at a time. On their papers, I have them find the distance from the front left and front right vertices to the top vertex.
We compare the answers and discuss them. Then we do the same for the back right and left vertices to the top vertex. This gives the students a chance to practice working the distance formula. For the end of the activity, I plot the point for the center of the base of the pyramid (-0.84, -1) and the students then find the distance from that point to the top vertex.
We then compare all of the distances and I bring up ideas of why these distances are different (camera angle in the photo, etc). Then I do some examples with figures on the smartboard that I have created that illustrate the more conventional 3D images that students see when finding the surface area and volume of pyramids.
I can send the smartboard document if you like via email.
After the Activity
I use a worksheet where students must find the distance of the slant height and perpendicular heights of several pyramids given coordinates on each pyramid.