Transcript 1-7 Three-Dimensional Figures

Solid Figures
Section 9.1
Goal:
• Identify and name solid figures.
Key Vocabulary
•
•
•
•
•
Solid
Polyhedron
Face
Edge
Base
Introduction to Three-Dimension
Figures
• Three-dimensional figures are
not flat figures. They have
length, width, and height.
They are also called solid
geometric figures.
Introduction to Three-Dimension
Figures
• A flat surface of a solid is a face.
• An edge is where two faces meet.
• A vertex is where three or more edges
meet.
• The face that is used to classify a solid is a
base.
face
edge
base
vertex
Types of Solids
• The surfaces of a three-dimensional
figure determine the type of solid it
is.
• A polyhedron is a threedimensional figure whose surfaces,
or faces, are all polygons.
• Prisms and pyramids are two
types of polyhedrons.
Properties of Polyhedra
• A polyhedron is a solid
bounded by polygons
called faces which enclose
a single region of space.
• An edge of a polyhedron
is a line segment formed by
the intersection of two
faces.
• A vertex of a polyhedron
is a point where three or
more edges meet.
• The plural of polyhedron is
either polyhedra or
polyhedrons.
Identifying Polyhedra
• Decide whether each of the solids below is a
polyhedron. If so, count the number of faces, vertices,
and edges of the polyhedron.
a.
b.
c.
This is a polyhedron. It has 5 faces, 6
vertices, and 9 edges.
This is not a polyhedron. Some of its faces
are not polygons.
This is a polyhedron. It has 7 faces, 7
vertices, and 12 edges.
Summary Types of Solids
The prism has two parallel congruent faces called bases. The other faces are
parallelograms… the pyramid has all but one of its faces (the base) intersecting at
one vertex… Both solids are named by the shape of their bases followed by “prism”
or “pyramid” as their surname (i.e. triangular prism and square pyramid).
Prisms
A prism is a polyhedron that has two parallel
congruent bases. The bases can be any
polygon. The other faces are parallelograms.
A cube is a special prism whose faces are all
congruent squares.
Cube
face
vertex
edge
A cube, a rectangular prism, has 6
faces (all squares).
Pyramids
A pyramid is a polyhedron that has one base.
The base can any polygon. The other faces are
triangles.
A regular tetrahedron is a special pyramid whose
faces are all congruent equilateral triangles.
Tetrahedron
A pyramid with four faces which are all
equilateral triangles.
Faces
4
Vertices
4
Edges
6
Polyhedrons or Polyhedra
(prisms and pyramids) are
named by the shapes of their
bases.
Triangular Prism
A prism with triangular parallel bases.
A triangular prism has five faces. Its base is a triangle.
Two of its faces are triangles; three of its faces are
rectangles. It has six vertices and nine edges.
Rectangular Prism
A prism with rectangular parallel bases.
Also called a Cuboid. A rectangular prism has six faces.
Its base is a rectangle. All of its faces are rectangles. It
has eight vertices and twelve edges.
Hexagonal Prism
A prism with hexagonal parallel
bases.
A hexagonal prism has eight faces.
Its base is a hexagon. It has six
faces that are rectangles and two
faces that are hexagons. It has
twelve vertices and eighteen edges.
Square (or rectangular) Pyramid
A shape with a square base and
triangular sides that meet at a point.
Vertices
5
Faces
5
Edges
8
Example: Naming Prisms and Pyramids
Identify the base or bases of the solid. Then
name the solid.
There are two bases, and they are
both octagons.
The other faces are parallelograms.
The figure is an octagonal prism.
Example: Naming Prisms and Pyramids
Identify the base or bases of the solid. Then
name the solid.
There is one base, and it is a
pentagon.
The other faces are triangles.
The figure is a pentagonal pyramid.
Try This: Example
Identify the base or bases of the solid. Then
name the solid.
There are two bases and they are
both triangles.
The other faces are parallelograms.
The figure is a triangular prism.
Try This: Example
Identify the base or bases of the solid. Then
name the solid.
All faces are congruent squares.
The figure is a cube.
More Practice
What is the name of a three-dimensional
figure with six square faces?
a.
b.
c.
d.
pyramid
cube
solid
edge
I am a three-dimensional figure with five faces. My
base is a rectangle; my other faces are triangles. I
have five vertices and eight edges. What am I?
a. rectangle prism
b. rectangular pyramid
c. triangular prism
d. triangular pyramid
How many edges does a square
pyramid have?
a.
b.
c.
d.
4
5
7
8
How many vertices does a
rectangular prism have?
a.
b.
c.
d.
12
8
6
4
Which of the following has a base
that is a triangle?
a.
b.
c.
d.
Triangular prism
Cube
Square pyramid
Rectangular pyramid
I am a three-dimensional figure with four faces.
All of my faces, including the base are
triangles. What am I?
a.
b.
c.
d.
rectangular prism
triangular prism
triangular pyramid
rectangular pyramid
How many faces of a cube are
squares?
a.
b.
c.
d.
2
4
6
8
Sally is going to build a table for her family room.
It is in the shape of a triangular prism. What
shapes will she need to build the table?
a. 5 triangles
b. 2 triangles and 3
rectangles
c. 2 triangles and 4
rectangles
d. 6 rectangles
Not Polyhedrons
• Other three-dimensional figures
include cylinders, cones, and
spheres.
• These figures are different from
polyhedrons because they each have
a curved surface and their bases are
not polygons.
• The curved surface of a cylinder or a
cone is called a lateral surface.
Cylinder
A prism with a circular cross-section.
The normal definitions of faces,
corners and edges are not
appropriate for a cylinder.
2 bases
A cylinder has two
parallel, congruent
circular bases connected
by a lateral surface.
Lateral surface
Cone
A cone has one circular base
and a lateral surface. The
lateral surface of a cone
comes to a point called its
vertex.
The point of the cone is directly
above the centre of the circular
base.
1 base Lateral surface
Vertex
Sphere
A sphere has only one
surface, which is curved,
and has no base. All of the
points on the surface are
the same distance from
the center of the sphere.
Top hemisphere
A plane that intercepts a
sphere through its center
divides the sphere into two
different halves, or
hemispheres.
Bottom hemisphere
Example
Determine whether the solid is a polyhedron. Then
identify the solid. If it is a polyhedron, name the
bases, faces, edges, and vertices.
Example
The solid has a curved surface, so it is not a
polyhedron. The base is a circle and there is
one vertex. So, it is a cone.
Answer: Base: circle T
Vertex: W
no faces or edges
Example
Identify the solid.
A. cone
B. cylinder
C. pyramid
D. polyhedron
Practice
Identify the type of each prism or pyramid.
1.
rectangular 2.
pyramid
pentagonal
prism
Identify the figure described.
3. two congruent circular faces connected by a curved
surface
cylinder
4. one circular face and a curved lateral surface that
comes to a point called a vertex
cone
Summary
Summary
Example 1
Identify and Name Polyhedra
Tell whether the solid is a polyhedron. If so, identify the
shape of the bases. Then name the solid.
a.
b.
SOLUTION
a.
The solid is formed by polygons so it is a
polyhedron. The bases are congruent triangles in
parallel planes. This figure is a triangular prism.
b.
A cylinder has a curved surface, so it is not a
polyhedron.
Example 2
Find Faces and Edges
Use the diagram at the right.
a. Name the polyhedron.
b. Count the number of faces and edges.
c. List any congruent faces and congruent edges.
SOLUTION
a. The polyhedron is a hexagonal pyramid.
b. The polyhedron has 7 faces and 12 edges.
c. Using the markings on the diagram, you can
conclude the following:
Example 2
Find Faces and Edges
Congruent faces
Congruent edges
∆PQV  ∆QRV  ∆RSV
PQ  QR  RS  ST  TU  UP
∆STV  ∆TUV  ∆UPV
PV  QV  RV  SV  TV  UV
Example 3
Sketch a Polyhedron
Sketch a triangular prism.
SOLUTION
1. Draw the triangular bases.
2. Connect the corresponding vertices of
the bases with vertical lines.
3. Partially erase the hidden lines to
create dashed lines. Shade the prism.
Your Turn:
Tell whether the solid is a polyhedron. If so, identify
the shape of the base(s). Then name the solid.
1.
ANSWER
yes; triangular;
triangular prism
ANSWER
no; cone
ANSWER
yes; rectangular;
rectangular pyramid
2.
3.
Your Turn:
4. Copy the partial drawing of a triangular
pyramid. Then complete the drawing of
the pyramid.
ANSWER
Euler’s Formula
• Mathematician
Leonhard Euler
proved that the
number of faces (F),
vertices (V), and
edges (E) of a
polyhedron are
related.
Leonard Euler
1707-1783
Using Euler’s formula
Use Euler’s Formula to find the number of edges on
a solid with 6 faces and 8 vertices.
F + V = E + 2Euler’s Formula
6 + 8 = E + 2 Substitute the number of faces and vertices.
12 = E
Simplify.
A solid with 6 faces and 8 vertices has 12 edges.
Regular Prisms and Regular Polyhedra
• If the bases of a prism are regular
polygons, then it is called a regular
prism.
• If all of the faces of a polyhedron are
regular congruent polygons and all of
the edges are congruent then it is called
a regular polyhedron.
Platonic Solids
• There are five (5) regular polyhedra called Platonic Solids, named
after the Greek mathematician and philosopher, Plato.
• The five Platonic Solids are:
1.
Tetrahedron (4 equilateral triangle faces)
Tetrahedron
2.
Hexahedron (6 square faces)
Hexahedron
3.
Octahedron (8 equilateral triangle faces)
Octahedron
4.
Dodecahedron (12 regular pentagon faces)
Dodecahedron
5.
Icosahedron (20 equilateral triangle faces)
Icosahedron
Summary
Review
Homework
• Pg. 476 – 480: #1 – 9 all, 11 – 23 odd, 25
– 30 odd, 31 – 67 odd