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

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### 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 = \( ^{\sqrt{a^{2}+b^{2}-2ab\cos c}} \)

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,

\(c^{2} = a^{2}+b^{2}-2ab\cos C\)

\( c^{2} = 10^{2}+12^{2}-2\left ( 10 \right )\left ( 12 \right )\cos C \)

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:

\( a\div \sin A= c\pm \div \sin C \) and \( b\div \sin B= c\pm \div \sin C \)

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,

\( 5\div \sin \left ( 30 \right )= b\div \sin \left ( 90 \right ) \)

\( b= 5\div \sin \left ( 30 \right ) \times \sin \left ( 90 \right ) \)

=\( 5\div.5 \times 1 \)

b= 10 inches.

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

\( 5\div \sin \left ( 30 \right ) = c\div \sin \left ( 60 \right ) \)

S\( o, c= 5\div.5 \times .87 \)

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.

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