Introduction :
Euler grew up near Basel, Switzerland, and studied at an early age
under Johann Bernoulli. He finished studies at the University of Basel
when only 15 years old. From 1727 to 1741, Euler worked in St.
Perersburg, Russia, and then moved to the Akademie in Berlin. In
1766 he returned to St. Petersburg, where he remained.
Euler's contribution to mathematics is represented here by a few of the
notations conventionalized by him or in his honor. Around the world,
these are read, written, and spoken thousands of times every day:
The Euler's Theorem, also known as the Euler's formula,
deals with the relative number of faces, edges and vertices
that a polyhedron (or polygon) may have. Let, for a given
polyhedron, F, E, V denote the number of faces, edges
and vertices, respectively.
Polyhedra :
A polyhedra (or planer body) is a solid or hollow body
that is made up entirely of plane faces and is three
dimensional. For example , a cuboid is a polyheron as it is entirely made up of planes. Planes
that make up cuboid are called its faces. There are six faces of the cuboid.
The plural of polyhedron is polyhedra. In Greek it means ‘many’ and hedra means ‘base’.
A polyhedra is named after its number of faces.
Each flat surface of a polyhedron is called a face and has a shape of polygon.
Each corner point of a polyhedron is called a vertex .
Two flat surfaces intersect in a line called an edge.
The solid shown is a triangular based pyramid or Tetrahedron. It has four vertices , 4 faces and
six edges. The tetrahedron is a simplest example of a solid for which all faces are polygons.
Some shapes and Names :
Names
Faces ( F )
Vertices ( V )
Edges ( E )
Tetrahedron
4
4
6
Heptahedron
7
7
12
Enneahedron
Octahedron
Triangular Prism
Pentagonal Prism
Shapes
TASK TO DO :
1. Complete the tables and write the values.
2. Look carefully at the columns and describe the pattern you observed in your own
words.
3. Generalise a mathematical relationship between Faces , Vertices and Edges.
4. Construct some 3-D figures by your own and check your formula/relation.
(Atleast 3 figures)
5. Using the above formula find the edges for the given figures.
6. Allegra Cole a carpenter, she designed a 3-D structure on her own . She took a wooden
cuboid of length 2 mts , breadth 1 mts and height 1 mts. She the cuts all the corners (as
shown in the figure) so as to make a face on each corners . Now find the number of
faces , vertices and the edges of newly formed figure and justify the expression you
deduced in Q.3 .
7. Justify the degree of accuracy of your findings.
8. What are the importance of your formula in context to real life.
9. Suggest some improvement of your methods you used.
10.Why Sphers and cylinders are not called Polyhedrons ?
RUBRICS
CRITERIA B: PATTERNS AND VARIABLES
LEVELS
0
1-2
3-4
5-6
7-8
LEVEL DESCRIPTORS

The student does not reach to the level described by any of the descriptors
below.
The student only
 Attempts to understand the problem select and apply a mathematical
problem solving technique.
 Attempts to identify the variables and describe the pattern between them.
 Attempts to workout general formula for the pattern observed between the
variables.
 Attempts to draw a conclusion and make predictions consistent with
findings.
 Attempts to justify the formula using logical arguments
The student is partially able to
 Understand the problem select and apply a mathematical problem solving
technique.

Identify variables and describe the patterns between them.

Workout a general formula for the pattern observed between the
variables.

Draw conclusion and make predictions consistent with findings.
 Justify the formula using logical arguments
The student is able to
 Understand the problem select and apply a mathematical problem solving
technique with few errors.

Identify variables and describe the pattern between them with few errors.

Workout general formula for the pattern observed between the variables
with few errors.

Draw conclusion and make predictions consistent with findings with few
errors.
 Justify the formula using logical arguments with few errors






The student is able to
Understand the problem completely, select and apply a suitable
mathematical problem solving technique.
Identify the variables and describe completely the patterns between them.
Workout a complete general formula for the pattern observed between
the variables.
Draw a conclusion correctly and make predictions consistent with
findings.
Justify the formulas completely using logical arguments.
CRITERIA D: REFLECTION IN MATHEMATICS
LEVELS
LEVEL DESCRIPTORS
0

The student does not reach to the level described by any of the
descriptors below.
1-2
The student is only attempts to able to
Explain why the results are correct and make sense in context to the
problem.
 Explain the importance of their findings.
 State and justify the degree of accuracy of their results where
appropriate.
 Suggests improvement to the methods wherever applicable.
The student is partially able to

3-4
Explain why the results are correct and make sense in context to the
problem.
 Explain the importance of their findings.
 State and justify the degree of accuracy of their results where
appropriate.
 Suggests improvement to the methods wherever applicable.
The student is correctly able to

5-6




Explain why the results are correct and make sense in context to the
problem.
Explain the importance of their findings.
State and justify the degree of accuracy of their results where
appropriate.
Suggests improvement to the methods wherever applicable.