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,
\( 5\div \sin \left ( 30 \right )= b\div \sin \left ( 90 \right ) \)
\( b= 5\div \sin \left ( 30 \right ) \times \sin \left ( 90 \right ) \)
=\(Â 5\div.5 \times 1 \)
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-Â 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.
I get a different answer for first example.
I got Q1 as 20.5
median 23 and
Q3 26