Geometry

Polyhedra

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Goals

Know terminology about solids. Identify solids by type. Use Euler’s Theorem to solve problems.

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Polyhedron

A solid that is bounded by polygons.

The polygons are faces.

An edge is the intersection of two faces.

A vertex is the intersection of three or more faces.

Face

Face

Fac

e

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Polyhedron

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Polyhedron Views

SolidWire Frame

Hidden Line

All three views will be used in these presentations, the text and other materials.

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Which of these are Polyhedrons?

YES

YESNO

NO YES

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Concave PolyhedraA diagonal, or part of a diagonal, is outside the figure.

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Regular Polyhedra

All of the faces are congruent, regular polygons.

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Cross Section

The intersection of a solid and a plane.

Cross section is a circle.

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Cross Section

What is the intersection now?

Cross section is a rectangle.

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What would the cross section be?

A Square

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Leonard Euler

1707 – 1783 Probably the greatest

mathematician of all time.

Worked in, and made enormous contributions to, every branch of mathematics.

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Euler’s Formula

Count F, the number of faces.

123456 F = 6

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Euler’s Formula

Count V, the number of vertices.

1

23

4

56

V = 8

78

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Euler’s Formula

Count E, the number of edges.

12

34

56

E = 12

78

9

1011

12

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Euler’s Formula

Faces =

6

Vertices =

8

Edges =

12

V + F = E + 2

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Euler’s Formula

Faces =

6

Vertices =

8

Edges =

12

6 + 8 = 12 + 2

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Euler’s Formula

Faces =

6

Vertices =

8

Edges =

12

6 + 8 = 12 + 2

14 = 14

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Euler’s Formula

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Try another figure…Faces =

Vertices =

Edges =

F + V = E + 2

5 + 5 = 8 + 2

10 = 10

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Euler’s Formula

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Solve:

A polyhedron has 8 faces and 12 vertices. How many edges does it have?

18 V + F = E + 2 12 + 8 = E + 2 20 = E + 2 E = 18

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Solve:

A polyhedron has 24 vertices and 36 edges. How many faces does it have?

14 V + F = E + 2 24 + F = 36 + 2 24 + F = 38 F = 14

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Solve:

A polyhedron has 32 faces and 60 edges. How many vertices does it have?

30 V + F = E + 2 V + 32 = 60 + 2 V + 32 = 62 V= 30

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The Platonic Solids

There are only five of them. They are regular, convex polyhedra. First described ca. 350 BC by Plato in

Timaeus. Have been found in many ancient cultures.

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The Five Platonic Solids

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Tetrahedron

Has four triangular sides.

Associated with fire.

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Hexahedron (cube)

Has six square sides.

Associated with earth.

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Octahedron

Has eight triangular sides.

Associated with air.

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Dodecahedron

Has 12 pentagonal faces.

Associated with the heavens.

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Icosahedron

Has 20 triangular faces.

Associated with water.

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Johannes Kepler In 1596 Kepler published a

tract called The Cosmic Mystery in which he envisioned the universe as consisting of nested Platonic Solids whose inscribed spheres determine the orbits of the planets, all enclosed in a sphere representing the outer heavens.

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Dungeons and Dragons

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Public Toilets in South KoreaThis is not a Platonic Solid. It is a compound polyhedron. Can you find out its correct name?

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Platonic Solid Links

Mathworld

GSP Icosahedron

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Summary

A polyhedron is a solid object. The sides are faces. Regular polyhedra have congruent faces. There are 5 regular polyhedra (the

Platonic Solids). Euler’s Formula: F + V = E + 2

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Homework