Regular Polyhedra and Polygons
A regular polygon is a polygon with all sides
congruent and all angles congruent such as
equilateral triangle, square, regular
pentagon, regular hexagon, ………..
Regular Polygons
9/11/2024
•
Polygons are simple closed
plane figures made with
three or more line segments.
•
Polygons cannot be made
with any curves.
•
Polygons are named
according to their number
of line segments, or sides.
Polyhedron
What is a polyhedron?
Circles are not
polygons
Identifying Polyhedra
A
polyhedron
is a solid that is bounded by
polygons, called
faces
, that enclose a single region
of space.
An
edge
of 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
Parts of a Polyhedron
Regular Polyhedra
A polyhedron is
regular
if all its faces are
congruent regular polygons.
regular
Not regular
Vertices are
not formed
by the same
number of
faces
3
faces
4
faces
A polyhedron is called a
regular polyhedron or Platonic Solids
if the
faces
of the polyhedron
are congruent regular polygonal regions
,
and
if each vertex is the intersection of the same number of edges
.
Polyhedra is the plural for polyhedron.
Regular Polyhedra
A polyhedron is called a
regular polyhedron or
Platonic Solid
if the
faces
of the polyhedron
are
congruent regular polygonal regions
, and
if
each vertex is the intersection of the same
number of edges
.
Polyhedra is the plural for polyhedron.
Relationship Between
Polygons and Polyhedra
A polyhedron and polygon share some of the same
qualities. A regular polyhedron’s face is the shape of a
regular polygon.
For example: A tetrahedron has a face that is an
equilateral triangle. This means that every face that
makes the tetrahedron is an equilateral triangle. Around
all the vertices and every edge is the same equilateral
triangle.
Relationship Between
Polygons and Polyhedrons
A polyhedron is made of a net which is basically like a
layout plan. It is flat and made of all the faces that
you will see on the polyhedron.
For example: A cube has six faces all of them are
squares. When you open the cube up and lay it out flat
you see all of the six squares that make up the cube.
Platonic Solids
Tetrahedron
Platonic Solids
Octahedron
Tetrahedron
Platonic Solids
Octahedron
Tetrahedron
Icosahedron
Platonic Solids
Cube
Octahedron
Tetrahedron
Icosahedron
Platonic Solids
Hexahedron
Octahedron
Dodecahedron
Tetrahedron
Icosahedron
~
~
There are only
There are only
five
five
platonic solids
platonic solids
~
~
,
Euler’s Formula and
Euler’s Formula and
Platonic Solids
Platonic Solids
https://www.youtube.com/watch?v=C36h00d7xGs&lis
t=PLW1zH15TmnByfL1TkOxJW-FuEcFvBH-
x_&index=10&t=0s
Leonhard Euler
•
Euler = “Oiler”
Regular polygons and polyhedra are fundamental shapes in geometry. A regular polygon has sides and angles that are congruent, such as equilateral triangles and squares. Polygons are simple closed figures made with line segments. Polyhedra are solids bounded by polygons as faces. Regular polyhedra are polyhedra with congruent regular polygon faces. They are also known as Platonic solids. The relationship between polygons and polyhedra is seen in how faces of regular polyhedra correspond to regular polygons. Nets of polyhedra show a flat layout plan of the faces.
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Presentation Transcript
Regular Polygons A regular polygon is a polygon with all sides congruent and all angles congruent such as equilateral triangle, pentagon, regular hexagon, .. square, regular
Polygons Polygons are simple closed plane figures made with three or more line segments. Polygons cannot be made with any curves. Polygons are named according to their number of line segments, or sides. 9/11/2024
Polyhedron What is a polyhedron? Circles are not polygons
Identifying Polyhedra A polyhedron is a solid that is bounded by polygons, called faces, that enclose a single region of space. An edge of 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
Regular Polyhedra A polyhedron is regular if all its faces are congruent regular polygons. Not regular 3 faces Vertices are not formed by the same number of faces 4 faces regular
Regular Polyhedra A polyhedron is called a regular polyhedron or Platonic Solid if the faces of the polyhedron are congruent regular polygonal regions, and if each vertex is the intersection of the same number of edges. Polyhedra is the plural for polyhedron.
Relationship Between Polygons and Polyhedra A polyhedron and polygon share some of the same qualities. A regular polyhedron s face is the shape of a regular polygon. For example: A tetrahedron has a face that is an equilateral triangle. This means that every face that makes the tetrahedron is an equilateral triangle. Around all the vertices and every edge is the same equilateral triangle.
Relationship Between Polygons and Polyhedrons A polyhedron is made of a net which is basically like a layout plan. It is flat and made of all the faces that you will see on the polyhedron. For example: A cube has six faces all of them are squares. When you open the cube up and lay it out flat you see all of the six squares that make up the cube.
Platonic Solids Tetrahedron
Platonic Solids Tetrahedron Octahedron
Platonic Solids Tetrahedron Octahedron Icosahedron
Platonic Solids Cube Tetrahedron Octahedron Icosahedron
Platonic Solids ~There are only five platonic solids~ Hexahedron Tetrahedron Octahedron Icosahedron Dodecahedron
, Euler s Formula and Platonic Solids https://www.youtube.com/watch?v=C36h00d7xGs&lis t=PLW1zH15TmnByfL1TkOxJW-FuEcFvBH- x_&index=10&t=0s V - E + F = 2
Leonhard Euler Euler = Oiler euler euler euler
