DRILLDRILL1)How many sides does

dodecagon have?

2)What type of triangle is this: Sides : 5, 8, 10?

TopicsTopics• Properties of PolyhedraProperties of Polyhedra

- Faces- Edges- Vertices

• NetsNets- Prisms- Pyramids- Cylinders- Cones

NetsNets

• A net is a two-dimensional figure that, when folded, forms a three-dimensional figure.

Identical NetsIdentical Nets

• Two nets are identical if they are congruent; that is, they are the same if you can rotate or flip one of them and it looks just like the other.

Nets for a CubeNets for a Cube

• A net for a cube can be drawn by tracing faces of a cube as it is rolled forward, backward, and sideways.

• Using centimeter grid paper (downloadable), draw all possible nets for a cube.

Nets for a CubeNets for a Cube

• There are a total of 11 distinct (different) nets for a cube.

Nets for a Rectangular PrismNets for a Rectangular Prism

• One net for the yellow rectangular prism is illustrated below. Roll a rectangular prism on a piece of paper or on centimeter grid paper and trace to create another net.

Another Possible Solution

• Are there others?

Nets for a Regular PyramidNets for a Regular Pyramid

• Regular pyramid– Tetrahedron - All faces are triangles

– Find the third net for a regular pyramid (tetrahedron)• Hint – Pattern block trapezoid and triangle

Nets for a Square PyramidNets for a Square Pyramid

• Square pyramid

– Pentahedron - Base is a square and faces are triangles

Nets for a Square PyramidNets for a Square Pyramid• Which of the following are nets of a

square pyramid?

• Are these nets distinct? • Are there other distinct nets? (No)

Nets for a CylinderNets for a Cylinder• Closed cylinder (top and bottom included)

– Rectangle and two congruent circles

– What relationship must exist between the rectangle and the circles?

– Are other nets possible?

• Open cylinder - Any rectangular piece of paper

Nets for a ConeNets for a Cone• Closed cone

(top or bottom included)– Circle and a sector of a larger

but related circle – Circumference of the (smaller)

circle must equal the length of the arc of the given sector (from the larger circle).

• Open cone (party hat or ice cream sugar cone)– Circular sector

Alike or Different?Alike or Different?

• Explain how cones and cylinders are alike and different.

• In what ways are right prisms and regular pyramids alike? different?

A polyhedron is a 3-dimensional figure whose surfaces are polygons.

The polygons are the faces of the polyhedron.

An edge is a segment that is the intersection of two faces.

A vertex is a point where

edges intersect.

Using properties of polyhedraUsing properties of polyhedra

• A polyhedron is a solid that is bounded by polygons called faces, that enclose a since region of space. An edge of a polyhedron is a line segment formed by the intersection of two faces.

Using properties of polyhedra

• A vertex of a polyhedron is a point where three or more edges meet. The plural of polyhedron is polyhedra or polyhedrons.

Ex. 1: Identifying Polyhedra

• Decide whether the solid is a polyhedron. If so, count the number of faces, vertices, and edges of the polyhedron.

a. This is a polyhedron. It has 5 faces, 6 vertices, and 9 edges.

b. This is not a polyhedron. Some of its faces are not polygons.

c. This is a polyhedron. It has 7 faces, 7 vertices, and 12 edges.

Types of Solids

Regular/Convex/Concave• A polyhedron is regular

if all its faces are congruent regular polygons. A polyhedron is convex if any two points on its surface can be connected by a segment that lies entirely inside or on the polyhedron.

continued . . .

• If this segment goes outside the polyhedron, then the polyhedron is said to be NON-CONVEX, OR CONCAVE.

Ex. 2: Classifying Polyhedra

• Is the octahedron convex? Is it regular?

It is convex and regular.

Ex. 2: Classifying Polyhedra

• Is the octahedron convex? Is it regular?

It is convex, but non- regular.

Ex. 2: Classifying Polyhedra

• Is the octahedron convex? Is it regular?

It is non-convex and non- regular.

Note:

• Imagine a plane slicing through a solid. The intersection of the plane and the solid is called a cross section. For instance, the diagram shows that the intersection of a plane and a sphere is a circle.

Ex. 3: Describing Cross Sections

• Describe the shape formed by the intersection of the plane and the cube.

This cross section is a square.

Ex. 3: Describing Cross Sections

• Describe the shape formed by the intersection of the plane and the cube.

This cross section is a pentagon.

Ex. 3: Describing Cross Sections• Describe the shape

formed by the intersection of the plane and the cube.

This cross section is a triangle.

• Polyhedron: a three-dimensional solid made up of plane faces. Poly=many Hedron=faces

• Prism: a polyhedron (geometric solid) with two parallel, same-size bases joined by 3 or more parallelogram-shaped sides.

• Tetrahedron: polyhedron with four faces (tetra=four, hedron=face).

Using Euler’s Theorem

• There are five (5) regular polyhedra called Platonic Solids after the Greek mathematician and philosopher Plato. The Platonic Solids are a regular tetrahedra;

Using Euler’s Theorem• A cube (6 faces)

• A regular octahedron (8 faces),

• dodecahedron

• icosahedron

Note . . .• Notice that the sum

of the number of faces and vertices is two more than the number of edges in the solids above. This result was proved by the Swiss mathematician Leonhard Euler.

Leonard Euler

1707-1783

Euler’s Theorem

• The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula

F + V = E + 2

Ex. 4: Using Euler’s Theorem

• The solid has 14 faces; 8 triangles and 6 octagons. How many vertices does the solid have?

Ex. 4: Using Euler’s Theorem

• On their own, 8 triangles and 6 octagons have 8(3) + 6(8), or 72 edges. In the solid, each side is shared by exactly two polygons. So the number of edges is one half of 72, or 36. Use Euler’s Theorem to find the number of vertices.

Ex. 4: Using Euler’s Theorem

F + V = E + 2

14 + V = 36 + 2

14 + V = 38 V = 24

Write Euler’s Thm.

Substitute values.

Simplify.Solve for V.

The solid has 24 vertices.

Ex. 5: Finding the Number of Edges• Chemistry. In molecules

of sodium chloride commonly known as table salt, chloride atoms are arranged like the vertices of regular octahedrons. In the crystal structure, the molecules share edges. How many sodium chloride molecules share the edges of one sodium chloride molecule?

Ex. 5: Finding the Number of Edges

To find the # of molecules that share edges with a given molecule, you need to know the # of edges of the molecule. You know that the molecules are shaped like regular octahedrons. So they each have 8 faces and 6 vertices. You can use Euler’s Theorem to find the number of edges as shown on the next slide.

Ex. 5: Finding the Number of Edges

F + V = E + 2

8 + 6 = E + 2

14 = E + 2 12 = E

Write Euler’s Thm.

Substitute values.

Simplify.Solve for E.

So, 12 other molecules share the edges of the given molecule.

Ex. 6: Finding the # of Vertices• SPORTS. A soccer ball resembles a polyhedron with 32 faces; 20 are regular hexagons and 12 are regular pentagons. How many vertices does this polyhedron have?

Ex. 6: Finding the # of Vertices• Each of the 20 hexagons has 6 sides

and each of the 12 pentagons has 5 sides. Each edge of the soccer ball is shared by two polygons. Thus the total # of edges is as follows.

E = ½ (6 • 20 + 5 • 12)

= ½ (180)

= 90

Expression for # of edges.

Simplify inside parentheses.

Multiply.

Knowing the # of edges, 90, and the # of faces, 32, you can then apply Euler’s Theorem to determine the # of vertices.

Apply Euler’s Theorem

F + V = E + 2

32 + V = 90 + 2

32 + V = 92 V = 60

Write Euler’s Thm.

Substitute values.

Simplify.Solve for V.

So, the polyhedron has 60 vertices.

Polyhedron # of Faces # of Vertices # of Edges

Cube 6 8 12Pyramid 5 5 8Figure #1 6 8 12Figure #2 4 4 6Figure #3 5 6 9

Homework

Pages: 304-305#’s 1-12, 14, 17-23