1. A Naturally Occurring Function
Euler’s Formula
A Naturally Occurring Function

2. Leonhard Euler was a brilliant Swiss mathematician
Leonhard Euler was a brilliant Swiss mathematician. He is often referred to as the “Beethoven of Mathematics.”
         

3. Euler discovered an interesting relationship between the number of faces, vertices, and edges for any polyhedron.

4. Poly-what?

5. A polyhedron is a 3 dimensional shape with flat sides.
                
A polyhedron is a 3 dimensional shape with flat sides.

6. All prisms and pyramids are examples of polyhedra (plural for polyhedron).
PRISMS PYRAMIDS

7. Any polyhedron has faces, vertices, and edges.
VERTEX

8. A face is a flat side.

9. This rectangular prism has 6 faces.

10. This rectangular prism has 6 faces.
FRONT

11. This rectangular prism has 6 faces.
BACK
FRONT

12. This rectangular prism has 6 faces.
TOP
BACK
FRONT

13. This rectangular prism has 6 faces.
TOP
BACK
FRONT
BOTTOM

14. This rectangular prism has 6 faces.
TOP
BACK
FRONT
LEFT
BOTTOM

15. This rectangular prism has 6 faces.
TOP
BACK
FRONT
RIGHT
LEFT
BOTTOM

16. This rectangular prism has 6 faces.
TOP
BACK
FRONT
RIGHT
LEFT
BOTTOM

17. This rectangular prism has 6 faces.
TOP
BACK
FRONT
RIGHT
LEFT
BOTTOM

18. This rectangular prism has 6 faces.
TOP
LEFT
BACK
FRONT
RIGHT
BOTTOM

19. This rectangular prism has 6 faces.
TOP
LEFT
BACK
FRONT
RIGHT
BOTTOM

20. This rectangular prism has 6 faces.
TOP
RIGHT
LEFT
BACK
FRONT
BOTTOM

21. This square pyramid has 5 faces.
The faces consist of 4 triangles and a square.

22. The faces consist of 4 triangles and a square.

23. A triangular pyramid has 4 faces.

24. A triangular pyramid has 4 faces.

25. A triangular pyramid has 4 faces.

26. A triangular pyramid has 4 faces.

27. A triangular pyramid has 4 faces.

28. A triangular pyramid has 4 faces.

29. An edge is a line segment where two faces meet.

30. A rectangular prism has 12 edges.

31. A triangular pyramid has 6 edges.

32. A vertex is a corner. It is a point that connects 2 or more edges.

33. A vertex is a fancy word for “corner.”B
Every triangle has 3 vertices (corners). Points A, B, and C are vertices.

34. A rectangular prism has 8 vertices.

35. A rectangular prism has 8 vertices.

36. A triangular pyramid has 4 vertices.

37. A triangular pyramid has 4 vertices.

38. Euler studied the faces, vertices, and edges of different polyhedra.

39. Like most great mathematicians and scientists, he organized his data in a chart.
Polyhedron
# of Faces
# of Vertices
# of Edges
Cube 6 8 12
Sq. Pyramid 5
Tri. Prism 9

40. Euler looked for a relationship between these numbers.
Polyhedron
# of Faces
# of Vertices
# of Edges
Cube 6 8 12
Sq. Pyramid 5
Tri. Prism 9

41. Can you determine Euler’s formula that relates the # of Faces and # of Vertices to the # of Edges?
Polyhedron
# of Faces
# of Vertices
# of Edges
Cube 6 8 12
Sq. Pyramid 5
Tri. Prism 9

42. Faces + Vertices –2 = Edges
Polyhedron
# of Faces
# of Vertices
# of Edges
Cube 6 8 12
Sq. Pyramid 5
Tri. Prism 9
- 2 = +
- 2 = +
- 2 = +

43. Use Euler’s Formula to determine the number of edges in a pentagonal prism.
Polyhedron
# of Faces
# of Vertices
# of Edges
Cube 6 8 12
Sq. Pyramid 5
Tri. Prism 9
Pent. Prism 7 10
- 2 = +
- 2 = +
- 2 = +

44. Use Euler’s Formula to determine the number of edges in a pentagonal prism.
Polyhedron
# of Faces
# of Vertices
# of Edges
Cube 6 8 12
Sq. Pyramid 5
Tri. Prism 9
Pent. Prism 7 10
- 2 = +
- 2 = +
- 2 = +
- 2 = +

45. Use Euler’s Formula to determine the number of edges in a pentagonal prism.
Polyhedron
# of Faces
# of Vertices
# of Edges
Cube 6 8 12
Sq. Pyramid 5
Tri. Prism 9
Pent. Prism 7 10 15
- 2 = +
- 2 = +
- 2 = +
- 2 = +

46. Euler’s Formula works for any polyhedron.
SUMMARY: Euler’s Formula says that if you add the number of faces and vertices, then subtract by 2, the result is the number of edges.
Euler’s Formula works for any polyhedron.

47. THE END!