Perimeter of a Triangle Formula

Definition of Perimeter

Any two-dimensional figure's perimeter is the distance measured around it. By adding the lengths of each of the sides, we can measure the diameter of any closed form. In this article, you will learn the definition of perimeter and how to calculate the perimeter of various types of triangles when all side lengths are defined. Students can find the diameter of a right triangle when only two of the side lengths are known. Finally, students will learn how to use the Law of Cosines to find the perimeter of any triangle for which they know two side lengths and the angle calculation between them (a "SAS Triangle"). 

The circumference of a triangle is the complete length of the triangle's outer boundary. Or, to put it another way, the triangle's circumference is equal to the number of its three sides. The perimeter is measured in the same way that the triangle's sides are measured.

If ABC is a triangle, where AB, BC and AC are the lengths of its sides, then the perimeter of a triangle formula ABC is given by:

Formula to find perimeter of triangle = AB + BC + AC

In addition, the circumference of a triangle has the same unit as the lengths of its edges. If the lengths of its sides are measured in separate units, convert them to the same unit first.

Triangles are divided into three groups based on the length of their sides:

  • Scalene Triangle

  • Isosceles Triangle

  • Equilateral Triangle


Perimeter of an Isosceles Triangle

Two equal sides and two equal internal angles define an isosceles triangle. If the base and sides of an isosceles are known, the diameter can be calculated.

Isosceles Triangle Perimeter Formula


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Perimeter of isosceles triangle = a + b + c

Since the sides AB and AC are of equal length, we may write the formula as follows.

Perimeter of isosceles triangle formula = a + a + c or  b + b + c

This can also be written as  2a + c or 2b + c or 2 x equal side + c


Semi perimeter of triangle

The Semiperimeter of the triangle is defined as the half of its perimeter. It is denoted by s.

The semi perimeter of triangle formula with side lengths a, b, and c is given by

\[s=\frac{a+b+c}{2}\]

The semi perimeter is used most often for triangles.


The Perimeter of an Equilateral Triangle Formula

An equilateral triangle is one with equal sides and an equal angle on both sides. The equilateral triangle's interior angles are almost the same, at 60 degrees.


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Where,

a is the side of an equilateral triangle.

h is the altitude of an equilateral triangle.

An equilateral triangle's perimeter is equal to a + b + c.

The length of the sides of an equilateral is equal.

As a result, the perimeter of an equilateral triangle is calculated as a + a + a= 3a.

This formula can also be written as:

The perimeter of an equilateral triangle = 3 x side


How to Find The Perimeter of A Triangle When You Do Know All Three Side Lengths?

When only two sides are given, we can measure the perimeter of a triangle in right triangles.


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The Pythagorean theorem is used for this. A right triangle is one that has one 90o angle. 

The triangle's legs, designated as a and b, form a 90o angle. The hypotenuse is the side of the triangle that separates the two legs and is opposite the 90o angle. The hypotenuse, or longest side of the triangle, is denoted by the letter c.

The following formulas can only be used to find the missing side of a right triangle when two other sides are known:

a2 + b2 = c2

\[a=\sqrt{c^{2}-b^{2}}\]

\[b=\sqrt{c^{2}-a^{2}}\]

\[c=\sqrt{a^{2}+b^{2}}\]

After we find the missing side, then we can use the perimeter formula to calculate the triangle's perimeter.  

When we know the lengths of two sides of a triangle as well as the angle that exists between them, we can use the Law of Cosines to find the missing side when we know side-angle-side (SAS).


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Each side of the triangle corresponds to the angle immediately opposite it when labelled this way. We can use the following formulas to find missing sides in any triangle given SAS information if we mark our triangle carefully:

a2 = b2 + c2 - 2bc.cosA

b2 = a2 + c2 - 2ac.cosB

c2 = a2 + b2 - 2ab.cosC

Once we've calculated the right side of the equation, we'll need to square both sides to get the final side length of the missing side. The perimeter of the triangle can be calculated using the perimeter formula.


Solved Examples:

1.Calculate the Circumference of a Triangle with 6 cm, 8 cm, and 12 cm Sides.

Ans: Given,

a = 6 cm

b = 8 cm

c =  12 cm

Formula of perimeter of a Triangle = a + b + c

= 6 + 8 + 12

= 26 cm


2. Calculate The Perimeter of An Equilateral Triangle With A 5 Cm Side.

Ans: We know that in an equilateral triangle, all the sides are equal.

Thus, a = 5

Formula of the perimeter of an equilateral triangle= 3a

= 3 x a

= 3 x 5

= 15 cm


3. Find The Perimeter of An Isosceles Triangle With 4 Cm Equivalent Sides And 7 Cm is The Length of The Third Side.

Ans: We know in an isosceles triangle, two sides are equal.

Given, a = 4, b = 4 and c = 7

Formula to calculate perimeter of an isosceles triangle = a +b + c = a + a + c = 2a + c

= 2 x 4 + 7

= 8 + 7

= 25 cm


Conclusion

We have discussed how to find the perimeter of a triangle when sides are known and also when any one of the sides is missing.

The perimeter of a triangle is defined in the following ways :

  • The cumulative length of a triangle's sides is known as its perimeter. The triangle's sides may be equal or different.

  • An equilateral triangle is one with the same length on all three sides.

  • If two of the sides are identical, the triangle is an isosceles triangle.

FAQs (Frequently Asked Questions)

1. What Does the Perimeter of a Triangle Mean?

Ans: The total distance around a triangle's edges is known as its circumference. In other words, the perimeter of a triangle is the length of its boundary.

2. How to Calculate the Perimeter of a Triangle if three Sides are Known?

Ans: To calculate the perimeter of a triangle, add the length of its sides. For instance, if a triangle has sides a, b, and c, its perimeter is P = a + b + c.

3. What is the Formula for the Perimeter of an Equilateral Triangle?

Ans: The formula for the perimeter of an equilateral triangle is 3a, where a is the side of the triangle.