M3 Authentic Work
Proving Euler’s Polyhedron Theorem
PENTAGON PRISM
BY
Napat Prasopsukh M39 #9
Tanat Ratanayatikul M39 #15
Jirapat Thanaratsuttikun M39 #21
Bhurinat Wangsutthitham M39 #23
M3: Authentic Work
Proving Euler’s Polyhedron Theorem
Introduction
This project is about Prism and to study the volume or the
all surface area of the specific prism. We use Google Sketch-up
8 to providing the picture to make it easier to understanding
than to only write the project. To understand the project you
must understand Euler Formula and 3 steps of Euler’s theorem
and also how it work or you will not understand anything + learn
anything from the project.
In geometry, a Platonic solid is a convex polyhedron that is
regular, in the sense of a regular polygon. Specifically, the faces
of a Platonic solid are congruent regular polygons, with the same
number of faces meeting at each vertex; although all its edges
are congruent, as are its vertices and angles.
Euler’s Formula
Euler Formula is V+F-E = 2. Where V = Vertex (Vertices)
or the angle of prism, F = Face or the area of prism and E =
Edge or the sides of the prism. For Example:
Cube: V = 8, F = 6 and E = 12; V+F-E=2 8+6-12 = 2
Euler’s Theorem Example
There are total 3-steps, and all the steps need the
understanding of 3D + 2D Geometry.
First of all we must deform the platonic solid by remove 1 face
and pull it down to the same level as the base.
Now it is 2D shape. And by Euler Theorem V = 8, F = 6 and E =
12; V+F-E=2 8+6-12 = 2 (The another face is Exterior face)
Step1: If there a face more than 3 sides, draw a diagonal line
splitting it.
Step2: Check if the network has faces which share only 1 edge
with the exterior face. If it does remove the face by removing
the shared edge one.
Step3: Check whether our network has a face which shares 2
edges with the exterior face. If it does, remove the face by
remove the edges and vertex.
Repeat Step 2&3
*** Note: If possible, do step 3 before step 2
Step4: Repeat Step 3
This is the end of prove. If you got the triangle, you did it
correctly but if don’t, you did it wrong or you didn’t complete it.
PS. It is 2-Dimensional but for easy to understanding so I make
it 3-Dimensional and this is example for the cube.
Euler Theorem Historical
It was Euler (presumably around 1740) who turned his
attention to the exponential function instead of logarithms, and
obtained the correct formula now coined after his name. It was
published in 1748, and his proof was based on the infinite series
of both sides being equal. Neither of these men saw the
geometrical interpretation of the formula: the view of complex
numbers as points in the complex plane arose only some 50
years later (see Caspar Wessel)
PENTAGON Prism Project Area
1) This is our platonic solid prism.
V+F-E=2 10+7-15=2
2) We remove the face and pull it out.
V+F-E=2 10+7-15=2
3) Then we pull all the edges down.
V+F-E=2 10+7-15=2 (face +1 by
exterior face, the removed one)
4) Use Euler’s Theorem Step1 by drawing the diagonal line to all
face with 3+ sides.
V+F-E=2
10+16-8=2
5) By Euler’s Theorem Step2 remove the face by remove edges.
V+F-E=2
10+9-17=2
6) By Euler’s Theorem Step3 remove the all faces that share 2
edges with exterior face.
V+F-E=2
5+4-7=2
7) After another Euler Theorem Step3 and we get the final
outcome: the triangle.
V+F-E=2 3+2-3=2
Project Conclusion
This project improves our understanding of the Geometry
shapes and prism. This make us to find the answer by ourselves
not to ask the teacher to get the answer easily and forget it
easily too. This will make us remember proves and theorem that
V+F-E=2 to use in the final exam. Also the project concerns the
picture parts that need us to participate on the Google Sketchup to build the Geometry shapes + prism so it make us learn in
many different way all together. For example: math, computing,
solving problems and logic of proves. Finally we’ve learn many of
these skills by this single project.