Euler's formula

Euler’s formula deals with shapes called Polyhedra.
  A Polyhedron is a closed solid shape which has flat faces and straight edges. 
An example of a polyhedron would be a cube, whereas a cylinder is not a polyhedron as it has curved edges.

1. Euler's Formula
Algebra
Prepared by
Yousef Elshrek

2. Euler's Formula

3. •Euler’s formula deals with shapes called Polyhedra.
•A Polyhedron is a closed solid shape which has flat faces and straight
edges.
•An example of a polyhedron would be a cube, whereas a cylinder is not
a polyhedron as it has curved edges.

4. •Euler’s formula states that for Polyhedra
that follow certain rules:
•F+V-E=2
•Where F= Number of faces, V= Number
of vertices (Corners), E=Number of
edges
•Let us check that this works for the
Tetrahedron (triangle-based pyramid)
and the cube.
Shape Faces Vertices Edges F+V-E
Tetrahedron 4 4 6 4+4-6=2
Cube 6 8 12 6+8-12=2

5. •Euler's Formula
•For any polyhedronthat doesn't intersect itself,the
•Number of Faces
•plus theNumber of Vertices(corner points)
•minus theNumber of Edges
•always equals 2
•This can be written:F + V − E = 2
Try it on the cube:
A cube has 6 Faces, 8 Vertices, and 12 Edges,
so: 6 + 8 − 12 = 2

6. •MississippiStandard:Constructthree-dimensionalfiguresusingmanipulatives
andgeneralizetherelationshipsamongvertices,faces,andedges(suchas
Euler’sFormula).
•Recallthatathree-dimensionalfigurehasalength,width,anddepth(or
height).Theflatsurfacesofathree-dimensionalfigurearethefaces.Theline
segmentswherethefacesmeetaretheedges.Thepointswheretheedges
intersectarethevertex.
•Thenumberoffaces,vertices,andedgesofathree-dimensionalfigureare
relatedbyEuler’sFormula.

7. Key Concept
Words In three-dimensional figure , the sum of the faces F and vertices V
is equal to two more than the number of edges E
Symbols F + V = E+ 2
You can verify Euler’s Formula for three-dimensional figures. For example, the
rectangular prism above has 6 faces, 8 vertices, and 12 edges.
F + V = E +2 Euler’s Formula
6 + 8 =12 + 2 Substitute 6 for F, 8 for V, and 12 for E.
14 = 14 ✓ Add. The sentence is true.
Since 14 = 14, the formula is true for the rectangular prism.

8. •Example
•Verify Euler’s Formula
1.Determine whether Euler’s Formula is true for the figure below.
•The figure has 5 faces, 6 vertices, and 9 edges.
•F+V = E +2 Euler’s Formula
•5 + 6 = 9 + 2 Substitute 5 for F, 6 for V, and 9 for E.
•11 = 11 ✓ Add. The sentence is true.
•Yes, the formula is true for the figure shown.

9. •Example
•USE EULER’S FORMULA
•A three-dimensional figure has 5 faces and 4 vertices. Use Euler’s
Formula to find the number of edges in the figure.
•Use Euler’s Formula and solve for E.
•F + V = E + 2 Euler’s Formula
•5 + 4 = E+2 Substitute 5 for F and 4 for V.
•9 = E + 2 Add 5 and 4.
•-2 = -2 Subtract 2 from each side.
•7 = E Simplify.