Geometry
Chapter 6, Polygons and Polyhedrons
In the following explorations, you'll investigate some unusual polygon- and polyhedron-shaped structures.
Part B, Exploring Polygons
1. Study the dimensions of the Pentagon building and traditional Navajo homes at the following Web sites.
a. Under "The Pentagon Building Proper," find the lengths of each side ("outer wall") of the Pentagon Building in Arlington, VA. Sketch the building showing its dimensions.
b. What is the perimeter of the Pentagon?
c. Find the sum of the interior angles of the Pentagon.
d. The interior angles of the Pentagon are congruent. What is the measure of each angle?
e. Read about hogans, the traditional homes of the Navajo Indians. Scroll down about halfway through the article to the photo of one of the "new-style" hogans that began to appear in the early 1900's. Sketch a new-style hogan with a maximum distance between vertices equal to the diameter of today's round hogans. Assume that all sides are congruent.f. If all the angles in the new-style hogan you have sketched are congruent, what is the measure of each angle?
g. What is the perimeter of the hogan you sketched? Explain your method.
h. Read about the Currier "Octagon House". Sketch the house, showing its dimensions.
i. Find the perimeter of the Currier house and the measure of each interior angle.
Part C, Exploring Polyhedrons
2. Compare the Washington Monument with ancient Egyptian obelisks.
a. Find the height of the Washington Monument. Round to the nearest foot.
b. Click on Egyptian obelisk in the first paragraph of the above article to learn about the mathematics of obelisks.c. Using the height h that you found in a, calculate the values of m, n, and p in the figure.
Use this sketch for c, d, and e.
d. Find the perimeter of the base of the
monument.
e. Find the number of faces, vertices, and edges in the obelisk. Then show that Euler's Formula holds for the structure.
