Honors Geometry Name ______Shapes and Solids unit

SHAPES AND SOLIDS

In this unit, we will study various two-dimensional shapes and three-dimensional solids. When we want to describe a solid, we will usually talk about the shapes of its faces, as well as how those faces are fit together to make a solid.

Nets of Cubes

Sometimes it’s useful to draw unfolded, flattened out pictures of solids. These diagrams are called nets. Let’s begin with a simple example: the six square faces of a cube.

Imagine unfolding a cube, so that its faces are laid out as a set of squares joined edge-to-edge. Theresulting flat shape is called a net of a cube.

Group activity: Investigate the following problem with the students at your table, but be sure to take notes in your own notebook.

The problem: Find all of the different possible nets of cubes.

You will need to reach an agreement with your group on how to interpret the word “different.” Write down your criteria for determining whether two nets are regarded as the same or not.

How can you be sure that you’ve found all the possible nets?

Nets of Cubes write-up

This assignment will be collected on Tuesday, December 3: Write and justify your final answer to the question, “What are all the different nets of cubes?” Besure to specify your criteria for what makes two nets “the same” or“different.” Give a convincing argument that shows without a doubt that you have found all the possible nets (e.g., asystematic search that exhausts all possibilities; think about using strategies and notation that were discussed in class).

Polyominos

A polyomino is a flat shape formed by joining squares edge-to-edge. (The main difference between a polyomino and a net is that a polyomino is not required to fold up into something.)

Let’s say that two polynominos are the same if they are rotated or reflected copies of each other.

Using two squares or using three squares, there aren’t very many different polyominos.

  1. The first interesting case involves polyominos made of 4 squares (called 4-polyominos). (Note: These are the shapes of the pieces in the computer game Tetris.)

a.Find all of the different 4-polyominos.

b.Check that your polyominos are truly different under the definition given above of what it means for two polyominos to be the same.

c.Write a convincing explanation of how you can be certain that you have found all of the possible 4-polyominos. Hint: Proceed systematically through all the possible configurations. Begin with polyominos that have 4 squares in a row: how many of these are there, and why? Then consider 4-polyominos with only 3 squares in a row: how are you sure that you’ve found all of these? And so on.

2.Repeat all the parts of problem 1 for polyominos made of 5 squares. There are quite a few of these. If you conduct your search in a systematic way from the start, it will be easier to write the justification in part c.

3.Of the 5-polyominos from problem 2, which of them can be folded into “boxes without lids”?

Polygons

A polygon is a two-dimensional shape formed by line segments connected endpoint-to-endpoint, with the segments intersecting only at their endpoints.

  • Names of parts of polygons: The segments are called sides and the points are called vertices.
  • Measurements associated with polygons: side lengths and angle measures.
  • n-gons: The term n-gon is an abbreviation for “polygon with n sides and n vertices.” Certain n-gons have special names, e.g.,3gon=triangle, 4-gon=quadrilateral, 5-gon=pentagon ,
    6gon =hexagon, 8-gonoctagon.
  • Regular polygons: A polygon is said to be a regular if it is both equilateral (all of its side lengths are equal) and equiangular (all of its angle measures are equal). Regular n-gons exist for every integer n 3.
  • The three angle measures of any triangle have a sum of 180°.

Problems

1.In polygons, why must the number of vertices and the number of sides bethe same?

2.Quadrilaterals (4-gons) that are both equilateral and equiangular are called squares.

a.Draw an example of a quadrilateral that is equilateral but not equiangular.

  1. Draw an example of a quadrilateral that is equiangular but not equilateral.

3.These questions will lead you to a formula for the sum of the angle measures in any polygon.

a.Show that the sum of the four angle measures in any quadrilateral must be 360°. Hint:Subdivide the quadrilateral into 2 triangles.

b.Show that the sum of the five angle measures in any pentagon must be 540°.

c.Find the sum of the six angle measures in any hexagon. Justify your answer.

  1. Write an algebraic formula (involving n) giving the sum of the n angle measures in any ngon. Explain how you get the formula.

4. Recall that in a regular n-gon, allof the angle measures are equal.

a.Write an algebraic formula (involving n) giving the measure of each angle in a regular ngon. Explain how you get the formula.

  1. Make a chart showing the values of the formula for all n between 3 and 12. (Example: for n = 4, the value should be 90°, meaning that all the angles of a square measure 90°).

n / no. of diagonals from a single vertex / total no. of diagonals
3
4
5
6
7
8
9
10
n

5.A diagonal of a polygon is any segment joining two vertices that is not a side of the polygon.

a.Make and fill in the chart at the right, showing each ngon from n = 3 up to 10.

b.For each column of the chart, write an algebraic formula that could be used to find the value for anyn.

More on diagonals of polygons

  1. If everyone in your class shook hands with everyone else, how many handshakes would that be? There are many different ways to approach this problem. Describe your way.

  2. Write your rule that enables you to find the number of diagonals in any regular polygon, given the number of sides.

3.Problem 2 asked about diagonals in regular polygons. Does your answer change ifyou consider irregular polygons like the concave hexagon shown below? Explain.


  • Look again at the pictures from Problem 2. In the pentagon, all of the diagonals intersect in pairs only. In the hexagon, three diagonals intersect at the center point. For which other regular polygons is there a point where more than two diagonals meet?
  • In a soccer tournament, 10 teams are competing. Each team must play every other team. How many games will there be?

A few more problems about nets

This assignment will be collected on Wednesday, December 4: Try these problems with nets, this time involving solids other than cubes.

  1. A square pyramid has a square base and triangular faces that meet at a vertex. Find anddraw all the nets for a square pyramid.
  1. Find and draw a net for a cylinder. Be sure to include the top and bottom circular faces.
  1. Below are four drawings of eight congruent triangles connected in some way. Decide which of these will fold up into a closed 3-dimensional figure and which will not. (A figure made from eight triangular faces is an octahedron.)

Cross sections of solids

A cross section is the face you get when you make one slice through an object. Problems 4-7 ask you to visualize cross sections of solid objects.

  1. What cross sections can you make by slicing a cube? Record which of the shapes below you are able to create. Describe how you did it..
  • a square
  • an equilateral triangle
  • a rectangle that is not a square
  • atriangle that is not equilateral
  • apentagon
  • ahexagon
  • an octagon
  • aparallelogram that is not a rectangle
  • atrapezoid
  1. Can you create any shapes that are not listed above? Draw and name any other

crosssections you can make.

  1. If you think any of the shapes on the list in problem 4 are impossible to make by

slicing a cube, explain what makes them impossible.

  1. What cross sections can you get from a
  • sphere?
  • cylinder?

Regular polyhedra

A polyhedron is a solid formed by polygons that enclose a single region of space. The plural ofpolyhedron is polyhedra.

  • Names of parts of polyhedra: The polygons are called faces, the segments are called edges, and the points are called vertices.
  • Regular polyhedra: If a polyhedron has regular polygon faces that are all identical, and if the faces meet at each vertex in exactly the same way, then it is called a regular polyhedron

Group activity: finding all of the regular polyhedra

It is possible to conduct an exhaustive search for regular polyhedra, so we can be certain that we’ve found them all. With the help of plastic models, fill in the chart below as completely as possible. Here is one possible approach you could use to be certain that you’ve found all the possibilities:

  • Find all the different regular polyhedra with regular triangle faces. List them on your chart. Justify that your answer is complete. Hint: First consider three faces coming together at each vertex, then four faces at each vertex, and so on. When can you stop looking?
  • What regular polyhedra exist with square faces? Justify your answer.

What about regular pentagons, regular hexagons, and generally regular n-gons as faces?

  • How many regular polyhedra did you find in all? Check your list against the classroom chart or.page 494of our text forpictures and names of the five regular types of polyhedra. They are often referred to as the Platonic Solids.

name of the regularpolyhedron / type of face; no. of sides or vertices ofeach face
(s) / no. of faces meeting at
each vertex
(n) / total of the measures of the angles that meet at each vertex / total number of faces
(f) / total number of edges
(e) / total number of vertices
(v)
reg. tetrahedron / triangle; 3 / 3 / 3 · 60° = 180° / 4 / 6 / 4

Homework on regular polyhedra

This is your homework due Thursday, December5.

  1. Refer to your regular polyhedron chart from the previous page. Do you see any patterns relating the numbers in the last three columns? Find two different equations involving the values f, e, and v. Hint: To find one of them, look at the sum f + v.
  2. Give geometric explanations for these equations relating the several columns of numbers in your regular polyhedron chart (see the chart headings for the meanings of s, n, f, e, and v). The first one has beenfor you.

a.Explain why fs = 2e.
Explanation: Each of the f faces has s sides. So before assembling the polyhedron, there are fs sides of faces. When putting the polyhedron together, each edge is formed by putting two of these sides together. Thus the number of edges must be .

b.Explain why fs = nv.

  1. Explain why nv = 2e. (You could just cite parts a and b thenuse the transitive property, but try making a separate geometric argument instead.)
  1. Complete the picture at right to show what the regular tetrahedron would look like if it were cut open along the three lateral edges and unfolded into one piece. One face is missing.
  1. Complete the picture of what the regular hexahedron (a cube) would look like ifit were cut open along the lateral edges and three top edges, then unfolded. Two faces are missing. How many other patterns are possible for a regular hexahedron?
  1. At right is a picture of what the regular icosahedron would look like if it were cut along some edges and unfolded into one piece. When folded back together, the five top triangles meet at one top point. The edge labeled x will line up with which edge: a, b, or c?
  1. The regular octahedron is similar to the icosahedron but has only eight equilateral triangles. Complete the picture at right to show what the octahedron would look like if it were cut along some edges and unfolded into one piece. Two faces are missing.
  1. The regular dodecahedron is made with 12 regular pentagons. Suppose you were to cut the dodecahedron into two equal parts. They would resemble two flowers, each having five pentagon-shaped petals around a center pentagon. If half of the dodecahedron were cut along edges connecting the petals and then unfolded, what would it look like? Complete the pattern for half a dodecahedron.

Semi-regular polyhedra

A semi-regular polyhedron is a polyhedron with the following properties:

  • The faces are regular polygons, but not all of the same type.
  • The same combination of polygons meets in the same order at every vertex.

For example, you could build a polyhedron with a mixture of regular triangle and square faces insuch a way that four polygons meet at every vertex, always arranged in the order
triangle-square-triangle-square. While the various semi-regular polyhedra have fancy names (theexample is known as a cuboctahedron), it is easier to describe them using numerical symbols showing what polygons meet in what order (in this case, 3.4.3.4).

The semi-regular polyhedra fall into three families.

Prisms: A semi-regular prism is a polyhedron with two regular n-gons that are joined by a set of squares. There is a semi-regular prism for every integer n ≥ 3.

Anti-prisms: A semi-regular anti-prism is a polyhedron with two regular n-gons are joined by a set of regular triangles. (The triangles are “alternating up and down.”) There is a semi-regular anti-prism for every integer n ≥ 3.

Archimedean solids: All other semi-regular polyhedra fall into this category. One example is the cuboctahedron mentioned above. It turns out that there are only 13 or 14 of these polyhedra (our classroom chart shows 13, but we may discuss the 14th variation).

This is your homework due Tuesday, December 10. Please do your work on two separate sheets (Prisms and Anti-prisms). I will be collecting one of them.

Prisms

1.Explain why the cube is considered to be part of the prism family.

2.A picture of the prism with two 6-gon faces appears on the preceding page. Count the faces, edges, and vertices.

3.Sketch a picture of the prism with two 7-gon faces. Count the faces, edges, and vertices.

4.For the prism with two 6-gon faces, the polygons meeting at each vertex can be described by the numerical symbol “6.4.4”. What is the numerical symbol for the prism with two 7-gon faces?

5.Generalize the results of the previous problems by answering the following questions about the prism with two n-gon faces. (Most of your answers will be formulas involving n.)

a.How many faces are there of each type?

b.What is the total number of faces?

c.How many edges are there?

  • How many vertices are there?
  • Does the relationship f – e + v = 2apply to prisms?
  • What is the numerical symbol describing this prism?

Anti-prisms

1.Explain why the octahedron is considered to be part of the anti-prism family.

2.A picture of the anti-prism with two 6-gon faces appears on the preceding page. Count the faces, edges, and vertices.

3.Sketch a picture of the anti-prism with two 7-gon faces. Count the faces, edges, and vertices.

4.For the anti-prism with two 6-gon faces, the polygons meeting at each vertex can be described by the numerical symbol “6.3.3.3”. What is the numerical symbol for the antiprism with two 7-gon faces?

5.Generalize the results of the previous problems by answering the following questions about the anti-prism with two n-gon faces. (Most of your answers will be formulas involving n.)

a.How many faces are there of each type?

b.What is the total number of faces?

c.How many edges are there?

d.How many vertices are there?

  1. Does the relationship f – e + v = 2apply to anti-prisms?

f.What is the numerical symbol describing this anti-prism?

Homework due WednesdayDecember 11: problems 1through 6.

Archimedean solids: beginning counting problems

Previously you studied some counting strategies for the faces, edges, and vertices of regular polyhedra. To do these problems, you must figure out how to modify those strategies to work for semi-regular polyhedra.

1.The picture at the right shows a semi-regular polyhedron called a truncatedicosidodecahedron. Its faces are as follows:

30 squares
20 hexagons
12 decagons (10-gons)

a.What is the numerical vertex code for this polyhedron?

b.How many edges are there?

c.How many vertices are there?

2.The picture at the right shows a net of the “4.6.8” polyhedron, the truncated cuboctahedron.

a.How many faces of each type are there?

b.How many edges are there? (Don’t try to count them!)

c.How many vertices are there? (Don’t try to count them!)