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Polyhedra

Polyhedra

Roll Over Plato
&
Tell Pythagoras the News...




A regular polyhedron is a solid with regular polygon surfaces. All of its faces, edges, and corners are identical. The five regular polyhedra are named after Plato who wrote a book about them titled Timaeus. He was not the first nor the last, however, to write about them. The Greeks believed the Cosmos consisted of 4 elements: Fire, Earth, Air, and Water.
  • The Tetrahedron represented Fire
  • The Hexahedron represented Earth
  • The Octahedron represented Air
  • The Icosahedron represented Water
  • The Dodecahedron represented the entire Cosmos
Today we know there are more than 4 elements, but we still categorize all matter into 4 states plus an unknown one:
  • Water is a Liquid
  • Air is a Gas
  • Earth is a Solid
  • Fire is a Plasma
  • Dark Matter is that mysterious something that pervades most of the Cosmos

In the contemporary world, Plato's Solids are used by Chemists to model molecules and atoms. Physicists use them to study crystals, minerals, and soap bubbles. Architects use them to model houses. Have you ever wondered why you live in a box rather than an icosahedron?


Dome


Printable cut-outs of all the Platonic solids may be found at Math Is Fun.
You may also like a printable cut-out of a Dodecahedral Desk Calendar.



TODAY'S QUESTION

Are there more than 5 Platonic Solids?


To answer this question, start by filling out the table below.
  1. POINTS: How many Vertices are on each polyhedra?
  2. LINES: How many Edges are on each polyhedra?
  3. AREAS: How many Faces are on each polyhedra?
  4. P+A-L: For each polyhedra, add the number of Points to the number of Areas and subtract the number of Lines
  5. SIDES/AREA: How many sides are on each polygon face?
  6. ANGLE@POINT: What is the angle made by the sides of each polygon?
    (Read below about how to find the angle of a regular polygon)
  7. AREAS@POINT: How many Faces meet at a Vertex?
  8. SUM@POINT: What is the sum of all the angles at a Vertex?




HOW TO FIND THE INTERIOR ANGLES
OF A REGULAR POLYGON

A polygon is an enclosed area with straight lines as boundaries. At each vertex of a polygon, the sides form an angle. These are called the interior angles. A polygon is regular if each of its interior angles are equal. The hexagon is a regular polygon. A polyhedron is regular if it has a regular polygon on each of its faces.
  1. The Tetrahedron has an equilateral triangle on each of its faces
  2. The Hexahedron cube has a square on each of its faces
  3. The Octahedron has an equilateral triangle on each of its faces
  4. The Dodecahedron has a pentagon on each of its faces
  5. The Icosahedron has an equilateral triangle on each of its faces




Notice that angle sum a+b in each of the polygons above equals 180 degrees. If we know angle b, we can find angle a by subtraction: a = 180° – b. Angle b, however, is the same as each of the central angles of a regular polygon, and angle a is the sum of the other two angles of a triangular slice: a = c + c. It's just got to be that way because all triangles on a flat surface add up to 180°.



The central angles of a regular polygon, like the hexagon, may be found by dividing 360° by the number of polygon sides. The central angles of a hexagon each equal b = 360°/6 = 60°. Since there are 180° in a triangle, the interior angles of a hexagon must equal a = c + c = 180° – 60° = 120°. Therefore, to find the interior angles of a regular polygon, use this formula: a = 180° – 360°/n, where n is the number of sides of the regular polygon. Use this formula to find the interior angles of the dodecahedron below. Now you can complete the last 3 columns in the table above.



QUESTIONS FOR YOUR IMAGINATION


  1. In the hexagon tiles above, what is the sum of the angles at each Vertex?
  2. Try to make a regular polyhedron with hexagons on each of its faces.
    What happens?
  3. Can the sum of the angles at the vertexes of a regular polyhedron equal 360°?
    What happens if it does?
  4. Can the sum of the angles at the vertexes of a regular polyhedron be more than 360°?
    What happens if it does?
  5. Imagine a regular polyhedron other than the five Platonic solids.
    1. How many Vertexes does it have?
    2. How many Edges does it have?
    3. How many Faces does it have?
    4. Add the number of Vertexes to the number of Faces and subtract the number of Edges?
      What number do you get?
    5. What is the polygon at each of its Faces?
    6. What are the degrees of the interior angles of each Face?
    7. How many Faces meet at a Vertex?
    8. What is the sum of the interior angles at each Vertex?
  6. Now answer Today's Question for 5 Points:
    Are there more than five Platonic solids? Why or why not?


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Last Modified 12/2/08 9:20 AM

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