Euler’S Formula and Perimeter Of Triangle Formula

Euler’s Formula

Euler’s formula is very simple but also very important in geometrical mathematics. It deals with the shapes called Polyhedron. A Polyhedron is a closed solid shape having flat faces and straight edges. For example, a polyhedron would be a cube but whereas a cylinder is not a polyhedron as it has curved edges. This Euler Characteristic will help us to classify the shapes. Let us learn the Euler’s Formula here.

Euler’s Formula

What is Euler’s Formula?

Before we go through the Euler’s formula, let’s know about the polyhedron shape in a bit more detail. A polyhedron is a solid object whose surface is made up of the flat faces which are bordered by the straight lines only. In it, each face is, in fact, a polygon. A polygon is a closed shape in the flat 2-dimensional plane which is made up of points joined by the straight lines.

 

Leonhard Euler (1707-1783) was a Swiss mathematician who was one of the greatest and most productive mathematicians of all time. He spent much of his career blind, but still, he was writing one paper per week, with the help of scribes. Euler gave one very popular formula called Euler’s Polyhedral formula.  Euler’s other formulae are in the field of complex numbers.

Euler’s formula states for polyhedron that these will follow certain rules:

F+V-E=2

Where,

F	Number of faces
V	Number of vertices (Corners)
E	Number of edges

Euler’s Formula for other shapes

Euler’s Formula does work only for a polyhedron with certain rules.  The rule is that the shape should not have any holes, and also it must not intersect itself. Also, it also cannot be made up of two pieces stuck together, like two cubes stuck together by one vertex.

If all of these rules are properly followed, then this formula will work for all polyhedron.  Thus this formula will work for most of the common polyhedral.

There are in fact many shapes which produce a different answer to the sum FE.  The answer to the sum FE is sometimes called the Euler Characteristic X.

This is often written as FE=X. Some shapes can even have a Euler Characteristic as negative value as for “Double Torus” surface.  So, it can start to get quite complicated value for complex figures.

This formula can be used in Graph theory. Such as:

1. To prove a given graph as a planer graph, this formula is applicable.
2. This formula is very useful to prove the connectivity of a graph.
3. To find out the minimum colors required to color a given map, with the distinct color of adjoining regions, it is used.

Solved Examples on Euler’s Formula

Q.1: For tetrahedron shape prove the Euler’s Formula.

Solution: In a tetrahedron,

FNumber of faces, F = 4

Number of Vertices, V= 4

Number of Edges, E = 6

Thus, F+V-E

= 4+4-6

= 2.

Hence proved.

Q.2: For cube shape prove the Euler’s Formula.

Solution: In a tetrahedron,

F = 6

V= 8

E = 12

Thus, F+V-E

= 6+8-12

=  2

Hence proved.

Perimeter of Triangle Formula

The term perimeter means a path that surrounds an area. It refers to the total length of the sides or edges of a polygon, a two-dimensional figure with angles. Let us learn the types of triangle and Perimeter of Triangle Formula.

The perimeter of Triangle Formula

                                                                                                                                                                     Source:pinterest.com

What is the Perimeter of the Triangle?

The result of the lengths of the sides is the perimeter of any polygon. In the case of a triangle:

Perimeter = Sum of the three sides.

The formula of Perimeter of a Triangle:

For a triangle to exist certain conditions need to be met the below conditions,

a+b> c

b+c> a

c+a> b

Hence, the formula for the Perimeter of a Triangle when all sides are given is,

P= a+b+c.

Where, a, b, c indicates the sides of the triangle.

One such example is when given sides are; a=6 cm, b=8 cm, c=5 cm. So we should add all the sides and hence the perimeter is 6+8+5= 19 cm.

Important Trigonometric derivations in finding the perimeter of a triangle, where;

Condition 1- when in a triangle we know (SAS)- Side Angle Side.

Use the law of cosines to find the third side and then the perimeter:

p = a2+b2−2abcosc√

Example say a triangle with side lengths 10 and 12, and an angle between them of 97°. We will assign variables as follows: a = 10, b = 12, C = 97°.

Now according to the formula,

c2=a2+b2−2abcosC

c2=102+122−2(10)(12)cosC

We can find the c from the above formula. Now we can easily calculate the perimiter of a trialgle using formula, P= a+b+c.

Condition 2- When in a triangle we know (ASA)- Angle Side Angle.

First, we have to find the third angle. As we know a triangle is a combination of 180 degrees total. So angle C is 180- angle A – angle B.

Use the law of sines to find remaining two sides and then the perimeter:

a÷sinA=c±÷sinC  and    b÷sinB=c±÷sinC

From the above formula we get all the sides now.

Hence, P = a+b+c.

Example- Imagine a triangle with sides a, b, and c, where the length of a = 5 inches. The two respective angles are 60 and degree. So, the third angle is 180 -60+90= 30 degree. Now using the law of Sines,

5÷sin(30)=b÷sin(90)

b=5÷sin(30)×sin(90)

=5÷.5×1

b= 10 inches.

We will do the same thing with side c, knowing that its opposite angle C is 60 degrees.

5÷sin(30)=c÷sin(60)

So,c=5÷.5×.87

c= 8.7 inch

Hence, the perimeter is 5+10+8.7= 23.5 inches.

Solved Examples on Perimeter of Triangle Formula

Q.1) Find the perimeter of a triangle whose sides are 3 cm, 5 cm, and 7cm

Ans-  According to the formula, P= a+b+c,

Hence, P = 3 + 5 + 7 = 15 cm.

Q. 2) If P = 30 cm and a = 5 and b = 7, what is c?

Ans- Using the formula P = a + b + c, replace everything given to you into the formula

Things that are given are P = 30, a = 8, and b = 10

Replacing them into the formula gives:

30 = 8 + 10 + c

30 = 18 + c

Hence, c = 12.