A. Volume of a Pyramid or Cone

Read the online page on Cones and Pyramids for tools and background. You can use those Facts in your explanations. Then put these pieces together. You have seen much of this in Lab 4.

1. Suppose C is a cone with base R, vertex V and height H. Then let R(d) be the intersection of the cone with the plane parallel to the base at distance d from the base (0<d<H). Explain why R(d) is similar to R and what is the ratio of similitude. (Hint: Use a dilation to transform R to R(d).)
2. If the area of R is A, what is the area of R(d). Notice that this area only depends on A, H and d. It does not depend on the shape of R or the location of V.
3. Conclude that if two cones (or pyramids) have the same base area and the same height, then they have the same area by a famous principle on the cone and pyramid page.
4. Prove that a pyramid whose base is a square of side 1 and whose height is 1 has a volume of 1/3. (Hint: You have constructed such a pyramid model and shown that 3 of them can be fit together to form a cube of side 1.)
5. Prove that a pyramid whose base is a square of side S and whose height is H has volume = (1/3)S²H.
6. Prove that any pyramid or cone whose base has area A and whose height is H has volume (1.3)AH. (Compare with a square pyramid of the same height and same base area.)

B. Dissections and Volumes

You can use persuasive arguments about shapes based on your experience in lab, modelmaking, etc. By a dissection of a polyhedron, we mean that the original polyhedron is broken up into polyhedral pieces that only overlap on faces.

1. Explain how a cube C can be dissected into a regular tetrahedron T whose vertices are 4 of the vertices of the cube. Notice that the remainder of the cube is made up of 4 other "corner" tetrahedra. If the edge length of the cube is s, use the volume formula for the corner tetrahedra to figure out the volume of the regular tetrahedron T. What is the ratio volume(T)/volume(C)?
2. Use problem 1 to answer this question: if the edge length of a regular tetrahedron is t, what is the volume? Then use your earlier work on the height of the regular tetrahedron to check that this volume is correct.
3. Build a regular tetrahedron T with side t from four regular tetrahedra U1, U2, U3, U4 with edge length t/2 and a regular octahedron O with edge length T/2. Use similarity to tell the ratio of volume(U1)/volume(T). Then use the dissection to find the volume of the octahedron. What is the ratio volume(octahedron)/volume(T)?
4. Now we have a nested picture: C constains T contains O. Notice that the vertices of O are the 6 midpoints of the sides of T which are in tern the centers of the 6 faces of C. Tell what is the ratio volume(O)/volume(C).
5. Finally, the 6 centers of the faces of O form the vertices of another cube C'. What is the ratio of volume(C')/volume(O) and also volume(C')/volume(C)?. (Suggestion: You may want to try coordinates or a model. The center of C can be the origin and the vertices of C can be taken as (1,1,1) etc. Then it is not hard to compute the vertices of C'.). Notice that C' is a dilation of C. What is the ratio of similitude?

C. Symmetries of a tetrahedron and cube (reporting experiments)

This set of problems is to report conclusions from experiments. It is possible to establish the same results from using coordinate, or factoring rotations into products of plane reflections, but all that is asked here is to carry out the experiments carefully.

1. List all the rotations of the tetrahedron by axis and angle. Then list all the plane reflections and rotoreflections.
2. You can keep track of a symmetry of a tetrahedron ABCD by telling what is the image of the vertices A, B, C, D. For example, the rotation by 120 degrees with axis D can be tabulated as below. It may help to experiment by labeling the vertices of your model tetrahedron and also labeling triangle ABC on a piece of paper. Then you can transform your model and track what label on the paper appears next to the label on the vertex. This is the talbe for T = rotation by 120 degrees with axis D.

Make similar tables for the following isometries:

• S = Rotation by 120 degrees with axis A
• M = Plane reflection in the perpendicular bisecting plane of AB
• H = Rotation by 180 degrees with axis through the midpoint of AB
• K = Rotation by 180 degrees with axis through the midpoint of AC
• R = Rotoreflection by angle 90 degrees with axis AB

3. Now experiment to answer the following questions. In each case tell in words which symmetry of the tetrahedron is the product and also include a table for your answer.

• What is the composition ST? Be careful to distinguish ST from TS.
• What is the composition HK?
• What is the composition MS?

4. Dihedral Angles

1. What is the dihedral angle between two faces of a regular octahedron? Since half of a regular octahedron is a square pyramid with equilateral triangle faces, what is the dihedral angle between the base and a face of the pyramid.
2. You have built a net for a square-base pyramid which is 1/3 of a cube. What are the dihedral angles of this pyramid?