Niki McCord
Math Presentation: Platonic Solids
Intro:
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Over 4,000 years ago, people started carving Platonic Solids in stone balls (which were found in
different places all over Scotland)
The Pyramids in Egypt were modeled after Platonic Solids as well
A Polyhedron is a solid with all flat surfaces that enclose a single region
Each flat surface is called a face
The line segments where the faces intersect are called edges
We have already learned about prisms, pyramids, cylinders, cones, and spheres; today we will
learn about five new polyhedrons called Platonic Solids.
Body:
 If all faces on a polyhedron are regular congruent polygons and all of the edges are congruent, it
is a regular polyhedron
 Regular polyhedrons are also known as Platonic Solids
 There are five Platonic solids
 Coined “Platonic Solids” because Plato (an ancient Greek teacher of Mathematics and
Philosophy) described them extensively in his writings
 According to Plato, the basic elements consist of small indivisible atoms that have the shapes of
Platonic solids
 Platonic Solids are: (use nets to show this)
o Tetrahedron: has 4 equilateral triangle faces, 3 triangles meet at each vertex
o Octahedron: has 8 equilateral triangle faces, 4 triangles meet at each vertex
o Icosahedron: has 20 equilateral triangle faces, 5 triangles meet at each vertex
o Hexahedron: has 6 square faces, 3 squares meet at each vertex
o Dodecahedron: has 12 pentagon faces, 3 pentagons meet at each vertex
 According to Plato:
o Tetrahedron symbolized dryness or fire because it has the least ratio of volume to surface
area
o Hexahedron symbolizes the stable earth because it rests solidly on its base
o Octahedron symbolizes air because it can rotate freely when grasped at opposite vertices
o Dodecahedron symbolizes the 12 signs of the zodiac because of it having 12 faces; it also
symbolizes the universe
o Icosahedron symbolizes water because it has the highest ratio of volume to surface area
 Why only five?
o Because when you add the internal angles that meet at a vertex (corner), it must be less
than 360 degrees
 If it were more than 360 degrees it would be curved and not a corner/point
o A regular (equilateral) triangle has an internal angle of 60 degrees, so we can have:
 3 triangles (3*60=180) Tetrahedron
 4 triangles (4*60=240) Octahedron
 5 triangles (5*60=300) Icosahedron
o A square has an internal angle of 90 degrees, so only 3 squares can meet at a vertex
(3*90=270) making a Cube or Hexahedron
Niki McCord
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A regular pentagon has internal angles of 108 degrees, so only 3 pentagons can meet at a
vertex (3*108=324) making a dodecahedron
Conclusion:
o According to Plato, the basic elements (fire, water, air, and earth) consist of small indivisible
atoms that have the shapes of the Platonic Solids
o A tetrahedron is made up of 4 equilateral triangles and looks like a pyramid
o A hexahedron is made up of 6 squares and resembles a cube
o An Octahedron is made up 8 equilateral triangles and looks like two square pyramids were put
together
o A dodecahedron is made up of 12 regular pentagons which makes it look like the inside of a bee’s
hive
o A icosahedron is made up of 20 equilateral triangles and is the platonic solid most resembling a
sphere
References:
Appel, R. (August 2012). Platonic solids. Retrieved October 25, 2012 from
http://www.3quarks.com/en/PlatonicSolids/index.html.
Boyd, C. J., Cummins, J., Malloy, C., Carter, J., & Flores, A. (2004). Geometry. New York, NY:
McGraw Hill Glencoe.
Platonic solids: Why five? (2012). Retrieved October 25, 2012 from
http://www.mathsisfun.com/geometry/platonic-solids-why-five.html.