Assignment 6A (Due Friday, 11/7)

6.1 Build a model of a square-based pyramid (10 points + bonus possible)

This problem 6.1 is to build a model of the pyramid introduced Monday in class. Read the instructions carefully so that you make the correct pyramid.

It is important that the size of the model is the 3-inch size specified below, since we will fit everyone's models together with others.

Let ABCD be a square face of a cube. Let the vertices A', B', C', D' be the vertices opposite A, B, C, D. Then our model will be a square-based pyramid with the vertex above one of the corners of the square (not in the center, as in the model we built with Polydrons). Specifically, the vertices of the pyramid are A, B, C, D, C', so the square ABCD is the horizontal base, the vertex C' is above A, and AC' = AB.

Assignment: With medium weight cardboard (i.e., index cards, manila folders but NOT paper), make a model of the square-based pyramid described above. Each edge of the square ABCD should be 3 INCHES in length. Tape the model neatly so that the model is reasonably accurate and craftspersonlike.

6.2 Theory of your model (20 points)

1. Exact edge length. If the edge length of the square ABCD is s, what are the exact, theoretical lengths of each of the 8 edges of the pyramid? Please write your answers as a multiple of s.
2. Decimal approximate edge length. Then set s = 3 and use a calculator to write down decimal approximations of the lengths in inches. Measure and check to see whether your answers are plausible.
3. Exact vertex angles of faces. On a piece of paper, draw accurately a net with each face of the pyramid. Label the edges with their lengths from (a). Compute the exact, theoretical value of the vertex angles (give an exact answer in degrees if there is one, but otherwise, use inverse trigonometric functions).
4. Decimal approximate vertex angles of faces.Use a calculator to give approximate decimal values for the angles in c, and measure to check plausibility.
5. Dihedral angles. For each of the 8 edges, find the exact, theoretical dihedral angle along that edge. Show your reasoning. (Of course there is repetition because some of these angles are congruent. You only need to work out each type once.)