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Objective: Students collect data and determine a relationship of different polygons.

Early mathematicians or philosophers would try and develop relationships by closely observing shapes. Leonhard Euler, a Swiss mathematician, was a sharp observer of solids.

As you try and figure out the relationship it is importance that the angles on each face of the polyhedron are what is important, not the angles between the faces.

Recreating the data and plotting the information and then graphing the data will help us determine how Euler came up with his formula. Use F = faces; V = vertex. Euler’s equation for determining the total measure of all the face angles is m=360(V-2) where m represents the total measure of all the face angles and V represents the number of vertex points. As you should have determined adding a vertex point in a polyhedron increases the total angle measure by 360 degrees. This is Euler’s equation.

The only restrictions on the figures were that there were no holes in them, that every face of a polyhedron had to be a polygon, and that all the faces were flat and edges straight.

Name	Shapes needed	# of faces	# faces at vertex	Total measure of face angles
Tetrahedron	4 eq. triangles	4	4	180x4=720
Hexahedron (cube)	6 squares	6	8	360 x 6=2160
Octahedron	8 eq. triangles	8	6	180 x 8=1440

Icosahedron	20 triangles	20	12	360x10=3600
Dodecahedron	12 pentagons	12	22	360x20=7200
Hexagonal prism	6 squares 2 hexagons	8	12	360x10=3600
Cuboctahedron	8 eq. triangles 6 squares	14	12	360x10=3600
Truncated tetrahedron	4 hexagons 4 eq. triangles	8	12	360x10=3600
Rectangular prism	2 squares 4 rectangles	6	8	360x6=2160
Rhombohedron	2 squares 2 rectangles 4 isosceles triangles	8	8	6x360=2160
Prismatic Hexagonal dipyramid	12 isosceles triangles 6 rectangles	18	14	12x360=4320