MEASUREMENT
Lesson 3

 

PHYSICAL SCIENCES Ė MEASUREMENT

LESSON 3.  Eulerís Formula for Polygons (Lab)      

 

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

 

Materials:

Polyconstructo shapes

            Angle ruler

 

Teacher Notes:

 

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. Eulerís 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 Eulerís 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.

ANSWERS:

 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

 

 

 

Vertices + Total Measure of Face Angles / 360  +2

 

It does not work.  These shapes have circular faces which do not have vertices

 

 

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