# Perimeter of a Triangle

Sub Topics
The perimeter can be defined as the total distance around the outside of a two dimensional figure. To measure the perimeter of a shape, one of the physical methods is to take a string, lay it along the outer edge of the shape until it reaches the start point of the string, and cut it off there. Straighten out the string and measure its length: that would give the perimeter of the shape. So, it is measured in units of length. But for larger figures, this will not be possible.

So the perimeter can be calculated by adding up together all the length of the sides of the figure provided. The formulas will differ depending on the figure taken.

## What is a Perimeter of a Triangle

Perimeter of triangle is nothing but the whole lenght of the triangle. It is taken as the sum of 2s = a + b + c. where s is the semi-perimeter.

## Formula for Perimeter of a Triangle

Given a triangle with side length a, b and c

Then perimeter, P = a + b + c.

This is true for all triangles.

For example:

Find the length of the given triangle:

So perimeter of the triangle = BC + AB + AC

= 4 units + 8 units + 7units

= 19units.

Hence the steps to be followed to find the perimeter of a triangle are:
1. Take the side measurement with similar units
2. Add all length of measurement.
3. Provide units.

Remember, to only add the terms when all the units are the same.

For example:

Find the perimeter of a triangle with sides of 9 inches, 7 inches and 1 foot.

It will be incorrect to write = 9 + 7 + 1 = 17 as units are different.

The correct way is = 9inches + 7inches + 1 foot = 16inches + 1 foot = 1 foot 16 inches

Or the whole thing can be converted to inches.

As 1 foot = 12 inches

So answer can be given as 2 foot 4 inches .

The general formula of perimeter of a triangle can be modified according to different types of triangles.

## How to Find the Perimeter of a Triangle

The perimeter of a triangle is given by 2s = a + b + c

where a = 1st side
b = 2nd side
c = 3rd side.

Lets mark three points, say E, F and G on a single plane.

Try to connect two points at a time, in all possible ways

The segments formed will be EF, FG and GE and it can be seen that a closed figure is formed. This is a triangle.

A simply closed figure obtained by joining three segments is called a triangle.

It is one of the basic shapes used in geometry.

Generally, it can be taken as a closed polygon with three corners joined together by three edges in the shape of a line segment. The corners are commonly called vertices and the edges as the sides of the triangles. Any triangle is named using the names denoted to the vertices.

For example, lets consider the ΔABC

Here the sides are AB, BC and CA each having its on length.

When perimeter of a triangle is taken, the sides are given more important than the angles of the triangle.

## Triangle Inequality Theorem

Any three measures have to fulfill certain inequality rule in order to be the measures of a triangle. Its called triangular inequality theorem.

Consider three measurement values; if any one value is shorter than the sum of the other two values, then these values corresponds to the three sides of a triangle. So, it means if any measurement values are greater than the sum of the others, it cannot form the sides of a triangle.

Example 1: Lets try to draw a triangle with the following measurement:

10, 5 and 2 units.

First draw the line say AB = 10 units

From A draw AC = 5units and from B draw BD = 2 units.

It can be seen that how much we try to draw, these lines wouldn’t meet.

Without trying to draw it this can be proved by using the inequality theorem.

As 10 + 2 = 12 > 5 units

10 + 5 = 15 > 10 units

But 5 + 2 = 7 < 10 units

So it doesn’t satisfy the inequality theorem.

Example 2: Lets try to draw a triangle with the following measurement:

3, 4 and 5 units

When we try to draw it,

It can be seen that the meet at the point C.

Also if we check the theorem, we get

4 + 3 = 7 > 5 units

4 + 5 = 9 > 3 units

3 + 5 = 8 > 7 units

Hence the points 3, 4, and 5 form the measures of a triangle.

Hence before finding the perimeter of a triangle, always keep in mind that the measurements given must satisfy the triangular inequality theorem.

## Equilateral Triangle

Here all sides are of equal measure.

Let a be the measure of the sides of an equilateral triangle.

Then its perimeter, P = a + a + a

P = 3a

## Isosceles Triangle

Here any of the two sides will be of equal measure.

Let a be the measure of the equal sides and c be the measure of the unequal sides of an isosceles triangle.

Then its perimeter, P = a + a + c

P = 2a + c

So in general, there are three formulas that can be used to find the perimeter of a triangle.

All sides are alike: equilateral P = 3a.

Two sides are alike: isosceles P = c + 2a.

No sides are alike: scalene  P = a + b + c.