Geometry
Chapter 6 Answers
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.
Answer:
b. What is the perimeter of the Pentagon?
[4,605 ft]c. Find the sum of the interior angles of the Pentagon.
[540°]d. The interior angles of the Pentagon are congruent. What is the measure of each angle?
[108°]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.
Answer:
f. If all the angles in the new-style hogan you have sketched are congruent, what is the measure of each angle?
[120°]g. What is the perimeter of the hogan you sketched? Explain your method.
Answer:
[The diagonals divide each other into 23 ÷ 2 = 12.5-ft segments. They also divide the hexagon into 6 equilateral triangles with sides of 12.5 ft. The perimeter is therefore
6 x 12.5 = 69 ft.]h. Read about the Currier "Octagon House". Sketch the house, showing its dimensions.
Answer:
i. Find the perimeter of the Currier house and the measure of each interior angle.
[106 ft; 135°]
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.
[555 ft]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.
[m = 55.5 ft; n = 55.5 ft; p = 499.5 ft]d. Find the perimeter of the base of the
monument.
[222 ft]e. Find the number of faces, vertices, and edges in the obelisk. Then show that Euler's Formula holds for the structure.
[F = 9, V = 9, E = 16; F + V E = 9 + 9 16 = 2]
Quadrilaterals are used in various types of artwork quite frequently. In this activity, you will see how quadrilaterals are used in decorative glass designs and how knowledge of the properties of various quadrilaterals can help artists create the desired shapes.
Part B, Proving Quadrilaterals Are Parallelograms
1. Look at stained glass designs by Frank Lloyd Wright.
a. Parallelograms play a major part in many of Wright's designs. Without actually checking that both pairs of opposite sides are parallel, how could an artist be sure that a quadrilateral is truly a parallelogram using only a ruler?
[Check that both pairs of opposite sides are congruent.]b. How could the artist be sure that the quadrilaterals are truly parallelograms using only a protractor?
[Check that both pairs of opposite angles are congruent or check that consecutive angles are supplementary.]
Part C, Proofs with Special Parallelograms
2. Look at some designs using Beveled
and textured glass. The first window you see has a square frame
with the diagonals creating part of a flower design. Describe the shape
of the first window with the flower design. What is true about the diagonals
for this shape?
[A rectangle; the diagonals are congruent.]
3. Peer through some amazing skylights at the following Web site.
a. Now look at a Skylight. The first window on this page has a regular octagonal frame. Each of the eight radiating sections of the octagon is composed of three quadrilaterals which appear to be trapezoids. Explain how the artist would know each of these shapes was a trapezoid if he or she knew that exactly two pairs of consecutive angles were supplementary.
[Extend one of the bases of the trapezoid and show that the two bases are parallel because the alternate interior (or exterior) angles are congruent. This means that the figure must either be a trapezoid or parallelogram. However, it cannot be a parallelogram because there are only two pairs of consecutive angles which are supplementary, and in a parallelogram, there would be four.]b. What are the measures of these pairs of supplementary angles?
[67.5° and 112.5°]
Regular polygons and polyhedrons occur frequently in nature. One place they occur is in the formation of molecules. Below, you will explore why some molecules take on the shape of one of the Platonic solids and how others are formed from a variation of these solids.
Part B, Regular Polyhedrons
1. Read a theory that predicts the Shapes of molecules. This theory concerns the number of electrons in the outermost principal energy level of an atom. It says that each atom in a molecule will have a shape which minimizes repulsion between electrons in the outermost principal energy level or valence shell of the atom. This means that the electrons in the valence shell are positioned to maximize the distance between each other. The positions of these valence electrons determine the shape of a molecule formed from the atoms. Two of the molecular shapes described in this article are the tetrahedron and the octahedron.
a. Methane (CH4) is a tetrahedral molecule. Which atoms form the vertices of the tetrahedron?
[The hydrogen atoms.]b. According to the VSEPR theory, why is the CH4 molecule shaped like a regular tetrahedron?
[There are four valence electrons on the carbon atom. Repulsion between pairs of atoms is minimized when they are distributed at the vertices of a regular tetrahedron. Hence, the C-H bonds form at these vertices.]
2. Refer to and page down the Shapes of molecules Web site for a picture of an octahedron.
a. Hexafluoride (SF6) takes on the shape of a regular octahedron. Which atoms form the vertices of the octahedron?
[The fluorine atoms.]b. What role does the sulfur atom play in forming this octahedron?
[There are six valence electrons on the sulfur atom. Repulsion between pairs of atoms is minimized when they are distributed at the vertices of a regular octahedron. Hence, the S-F bonds form at these vertices.]
3. Learn about a newly discovered carbon molecule called the "Buckyball," a nickname for "Buckminsterfullerine." You can build your own Buckyball at this Web site.
a. How many vertices does a Buckyball have?
[60]b. What are at the vertices of the Buckyball?
[Carbon atoms.]c. Explain how the Buckyball can be formed from a regular icosahedron. Click on "Figure 1" to see a picture showing how to truncate the icosahedron.
[The icosahedron has 12 vertices with 5 faces meeting at each vertex. To form a Buckyball, cut off a pentagonal cone one-third of the way along each of the five edges at each of the 12 vertices.]d. Explain why truncating the icosahedron in this manner creates a polyhedron with 60 vertices.
[The icosahedron has 12 vertices. However, after truncating in the manner described above, each of these vertices is replaced by a pentagon. So, for every vertex in the icosahedron, there are 5 vertices in the Buckyball. Thus, the Buckyball has 60 vertices.]