14.3 Geometry in Three Dimensions
Definition
Simple
Definition
Simple
Closed
Definition
Simple
Closed
Definition
A simple closed surface has exactly one interior, no holes and is
hollow.
Definition
Simple
Closed
Definition
A simple closed surface has exactly one interior, no holes and is
hollow.
Definition
A solid is a simple closed surface taken with its interior.
Definition
Simple
Closed
Definition
A simple closed surface has exactly one interior, no holes and is
hollow.
Definition
A solid is a simple closed surface taken with its interior.
Definition
A polyhedron is a solid where each face is a polygon.
Parts of a Polyhedron
Naming Polyhedron
Shape of base
Whether or not the faces are perpendicular to the base (Right)
Type of polyhedron
Nets
Definition
A net is a two dimensional representation of a three dimensional
object. Nets are not unique with respect to the orientation of the faces
of the figure.
Prisms
Prisms
Properties
two bases, which are necessarily parallel
Prisms
Properties
two bases, which are necessarily parallel
lateral faces are all parallelograms
Prisms
Properties
two bases, which are necessarily parallel
lateral faces are all parallelograms
if a right prism, the bases are perpendicular to the lateral faces
Prisms
Properties
two bases, which are necessarily parallel
lateral faces are all parallelograms
if a right prism, the bases are perpendicular to the lateral faces
if a right prism, the lateral faces are rectangles
Pyramids
Pyramids
Properties
one base
Pyramids
Properties
one base
lateral faces are all triangles, meeting at the apex
Pyramids
Properties
one base
lateral faces are all triangles, meeting at the apex
if a right pyramid, the altitude is perpendicular to the center of
the base
Pyramids
Properties
one base
lateral faces are all triangles, meeting at the apex
if a right pyramid, the altitude is perpendicular to the center of
the base
if a right pyramid, the lateral faces are isosceles triangles
Cylinders
Cylinders
Properties
two bases that are necessarily parallel but not necessarily circles
Cylinders
Properties
two bases that are necessarily parallel but not necessarily circles
one lateral face, which when ‘opened up’ is a parallelogram
Cylinders
Properties
two bases that are necessarily parallel but not necessarily circles
one lateral face, which when ‘opened up’ is a parallelogram
if bases are directly over each other, meaning that the lateral face
is perpendicular to the bases, then we have a right cylinder
Cylinders
Properties
two bases that are necessarily parallel but not necessarily circles
one lateral face, which when ‘opened up’ is a parallelogram
if bases are directly over each other, meaning that the lateral face
is perpendicular to the bases, then we have a right cylinder
if a right cylinder, the lateral face is a rectangle
Cylinders
Properties
two bases that are necessarily parallel but not necessarily circles
one lateral face, which when ‘opened up’ is a parallelogram
if bases are directly over each other, meaning that the lateral face
is perpendicular to the bases, then we have a right cylinder
if a right cylinder, the lateral face is a rectangle
Note: If not right, then the figure is called oblique.
Cones
Cones
Properties
one base, not necessarily a circle
Cones
Properties
one base, not necessarily a circle
one lateral face, which when ‘opened up’ is a portion of a circle
Cones
Properties
one base, not necessarily a circle
one lateral face, which when ‘opened up’ is a portion of a circle
the point at the top is the apex
Cones
Properties
one base, not necessarily a circle
one lateral face, which when ‘opened up’ is a portion of a circle
the point at the top is the apex
if the altitude is perpendicular to the base, we have right cone
Spheres
Spheres
Properties
one face
Spheres
Properties
one face
no edges
Spheres
Properties
one face
no edges
every point on sphere is equidistant from the center
Spheres
Properties
one face
no edges
every point on sphere is equidistant from the center
gray circle is called the great circle
Euler’s Formula
There is a relationship between the number of vertices, faces and
edges in any simple polyhedron. The shape of the base doesn’t matter
- the formula will hold.
Using your nets for polyhedra, count the number of faces, edges and
vertices and look for a relationship.
Euler’s Formula
There is a relationship between the number of vertices, faces and
edges in any simple polyhedron. The shape of the base doesn’t matter
- the formula will hold.
Using your nets for polyhedra, count the number of faces, edges and
vertices and look for a relationship.
Euler’s Formula
Euler’s Formula
For a simple polyhedra, F + V = E + 2.
Platonic Solids
Definition
A Platonic solid is a regular, convex polyhedron with all faces
congruent and the same number of faces meeting at each vertex.
Only five polyhedra are knows to satisfy this definition.
Tetrahedron
Hexahedron
Octahedron
Dodecahedron
Icosahedron
Example
Example
What is the length of the longest rod we can fit in a right circular
cylinder with radius 3 inches and height 8 inches?
Example
Example
What is the length of the longest rod we can fit in a right circular
cylinder with radius 3 inches and height 8 inches?
3 in
8 in
Solution
3 in
3 in
D
8 in
So what we have is a right triangle with legs 6 and 8 inches long. So,
by the Pythagorean theorem, we have
62 + 82 = D2 ⇒ D2 = 100 ⇒ D = 10
So, the length of the longest rod is 10 inches.