Perimeter of a Triangle Formula

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Perimeter of Triangle

Perimeter of Triangle: The perimeter of any two-dimensional figure is defined as the distance around the figure. We can calculate the perimeter of any closed shape just adding up the length of each of the sides. In this article, you will first learn about what is the perimeter, how to find the perimeter of different types of triangles when all side lengths are known. Furthermore, the solved examples will help you to get more views on the topic.

What is the Perimeter of a Triangle?

The sum of the lengths of the sides is the perimeter of any polygon. In the case of a triangle,

Perimeter = Sum of the three sides

Always include units in the final answer. If the sides of the triangle are measured in centimetres, then the final answer should also be in centimetres.

Perimeter of Triangle Formula

The formula for the perimeter of a closed shape figure is usually equal to the length of the outer line of the figure. Therefore, in the case of a triangle, the perimeter will be the sum of all the three sides. If a triangle has three sides a, b and c, then,

Perimeter, P = a + b +c

Perimeter of an Isosceles, Equilateral and Scalene Triangle

Below table helps us to understand how to find the perimeter of different triangles- Equilateral triangle, Isosceles triangle and Scalene triangle.

Where a, b, c and l are the side lengths and P = Perimeter.

This formula implies to find the perimeter of a triangle, add the lengths of all of its 3 sides together. If A, B and C are the side measures, and X is perimeter then

Perimeter of Right Triangle

A right triangle has a base(b), hypotenuse(h) and perpendicular(p) as its sides, By the Pythagoras theorem, we know,

h2 = b2 + p2

Therefore, the Perimeter of a right angle triangle= b + p + h

Examples

Let us consider some of the examples on the perimeter of a triangle:

Example 1: Find the perimeter of a polygon whose sides are 5 cm, 4 cm and 2 cm.

Solution: Let,

a = 5 cm

b = 4 cm

c = 2 cm

Perimeter = Sum of all sides = a + b + c = 5 + 4 + 2 = 11

Therefore, the answer is 11 cm.

Example 2: Find the perimeter of a triangle whose each side is 10 cm.

Solution: Since all three sides are equal in length, the triangle is an equilateral triangle.

i.e. a = b = c = 10 cm

Perimeter = a + b + c

= 10 + 10 + 10

= 30

Perimeter = 30 cm.

Example 3: What is the missing side length of a triangle whose perimeter is 40 cm and two sides are 10 cm each?

Solution: Given,

Perimeter = 40 cm

Length of two sides is the same i.e. 10 cm.

Thus, the triangle is an isosceles triangle.

Using formula: P = 2l + b

40 = 2 * 10 + b

40 = 20 + b

or b = 20

Missing side length is 20 cm.

What does the Perimeter of a Triangle Mean?

The perimeter of a triangle is the total distance around the edges of a triangle. In other words, the length of the boundary of a triangle is its perimeter.

How to Calculate the Perimeter of a Triangle?

To calculate the perimeter of a triangle, add the length of its sides. For example, if a triangle has sides a, b, and c, then the perimeter of that triangle will be P = a + b + c.

Calculate the Perimeter of a Right Triangle with Base as 3 cm and height as 4 cm.

First, using the Pythagorean theorem, calculate the hypotenuse of the right triangle.

h =√(base2+perpendicular2)

h = √(32+42)

h = √(9 + 16)

h = √25

Or, h = 5 cm

So, the perimeter of the triangle = 3 + 4 + 5 = 12 cm.

Perimeter of a Triangle

The perimeter of a triangle is defined as the total length of its boundary. A triangle is a polygon with 3 sides and it can be classified into different types based on the measure of its sides and angles. There are different formulas and methods to calculate the perimeter of a triangle based on the type of triangle.

What is the Perimeter of a Triangle?

The perimeter of a triangle means the sum of all three sides. The word perimeter is a combination of two Greek words – “peri” which means around and “metron” which means measure. The total distance around any 2D shape is defined as its perimeter. Since perimeter gives the length of the boundary of a shape, it is expressed in linear units.

Real-Life Example of Triangle’s Perimeter: Imagine that we need to fence the triangular park shown below. Now, to know the dimensions of the fence, we add the lengths of the three sides of the park. This length or distance of the boundary of a triangle is called the perimeter of the triangle.

Perimeter of a Triangle Formula

To calculate the perimeter of a triangle, we simply add the lengths of the sides given. The basic formula used to calculate the perimeter of a triangle is:

Perimeter = sum of the three sides

Let us understand this formula with the different types of triangles.

Perimeter of a Scalene Triangle

If a triangle has all three sides of different lengths, it is a scalene triangle. The perimeter of a scalene triangle can be calculated by finding the sum of all the unequal sides. The formula for the perimeter of a scalene triangle is Perimeter = a + b + c, where “a”, “b”, and “c” are the three different sides.

Perimeter of an Isosceles Triangle

If a triangle has two sides of equal length, it is an isosceles triangle. The perimeter of an isosceles triangle can be calculated by finding the sum of the equal and unequal sides. The formula for the perimeter of an isosceles triangle is: Perimeter of an isosceles triangle = 2a + b units.
where,

• a = sides of equal length
• b = the third side

Perimeter of an Equilateral Triangle

An equilateral triangle has all the sides of equal measure. The formula for the perimeter of an equilateral triangle is:
Perimeter of an equilateral triangle = (3 × a) units.
where ‘a’ = length of each side of the triangle.

Perimeter of a Right Triangle

A triangle that has one of the angles as 90° is called a right-angled triangle or a right triangle. The perimeter of a right triangle can be calculated by adding the given sides. The formula to calculate the perimeter of a right triangle is:

Perimeter of a right triangle, P = a + b + c units.

Since this is a right triangle, we can use the Pythagoras theorem, if any one side of this triangle is not known. The Pythagoras theorem says that the square of the hypotenuse is equal to the sum of squares of the other two sides. Referring to the figure given above:

• a = Perpendicular
• b = Base
• c = Hypotenuse of the right triangle

Hence, according to the Pythagoras theorem, c2 = a2 + b2. In this case, the perimeter of a right triangle can also be written as: P = a + b + √(a2 + b2). This is because c2 = a2 + b2 , therefore, c = √(a2 + b2).

Perimeter of Isosceles Right Triangle

A right triangle with two equal sides and two equal angles is called an isosceles right triangle. The perimeter of an isosceles right triangle can be calculated by adding the given sides.

The formula to calculate the perimeter of an isosceles right triangle is P = 2l + h, where l is the length of two equal legs or sides of the triangle, and h is the hypotenuse.

Another interesting point to be noted here is that using the Pythagoras theorem, we know, h = √(l2+ l2) = √2 × l or, l = h/√2 units. Therefore, the perimeter of an isosceles right triangle can also be written as: P = 2l + (√2)l = (2 + √2)l units.

Also, P = 2(h/√2) + h = (√2 × h) + h units.

How to Find The Perimeter of a Triangle?

The perimeter of a triangle can be calculated by following the steps given below:

• Step 1: Note the measurements of all the sides of a triangle and check that all the sides should have the same unit.
• Step 2: Calculate the sum of all the sides.
• Step 3: Give the answer along with the unit.

Let us see how to find the perimeter of a triangle using an example.

Example: Find the perimeter of △ABC having the following dimensions: AB = 6 inches, BC = 8 inches, AC = 10 inches.

Solution:

Step 1: Check if all three sides of the triangle are known.

AB = 6 inches, BC = 8 inches, AC = 10 inches

Step 2: Use the appropriate formula and add the sides to get the perimeter. Since this is a scalene triangle, we use the formula, Perimeter = a + b + c. Write the perimeter along with its units.

Perimeter of triangle ABC = 6 + 8 + 10 = 24 inches.

Perimeter of a Triangle Examples

Example 2: Find the length of the missing side of a triangular-shaped road sign whose perimeter is 48 inches and the two sides are 17 inches each.

Solution:

Let the length of the missing side be b.

Given, Perimeter = 48 inches

Length of the two equal sides = 17 inches each

Perimeter of a triangle = sum of lengths of three sides

48 = 17 + 17 + b

48 = 34 + b

b = 14

Therefore, b = 14 inches

Answer: Length of the missing side = 14 inches.

Example 3: The perimeter of a rectangular wire is 297 inches. The same wire is bent into the shape of an equilateral triangle. Find the length of each of its sides.

Solution:

We know that, the perimeter of a rectangle = total length of the wire

Length of the wire used = Perimeter of the triangle formed

Perimeter of an equilateral triangle = 3 × a

297 = 3 × a

a = 99

Answer: The length of each side of the triangle = 99 inches

FAQs on Perimeter of Triangle

What Is the Perimeter of a Triangle in Math?

The perimeter of a triangle is defined as the total length of its boundary. It is the sum of all three sides of the triangle.

What Is the Formula of Perimeter of Triangle?

The perimeter of a triangle can be calculated by simply adding the length of all the sides. The basic formula to calculate the perimeter of a triangle with sides ‘a’, ‘b’, and ‘c’ is a + b + c.

How Do You Find the Perimeter of a Triangle With Three Equal Sides?

To calculate the perimeter of a triangle with three equal sides, we add the length of all sides or multiply the length of any one side by 3. Such a triangle is called an equilateral triangle. The formula to calculate the perimeter of an equilateral triangle is 3a, where ‘a’ is the length of each side.

Can a Triangle Have the Same Area and Perimeter?

A triangle can have the same perimeter and area only in some special cases. These shapes having an equal perimeter and area are called equable shapes. Thus, a triangle with an equal perimeter and area is called an equable triangle.

How Do You Find the Third Side and Perimeter of a Right Triangle Given Two Sides?

The third side of a right-angled triangle can be calculated using the measure of the other two sides, by applying the Pythagoras theorem. According to the Pythagoras theorem, for any right-triangle with sides ‘a’, ‘b’, and ‘c’,
c2 = a2 + b2
where,

• a = Perpendicular
• b = Base
• c = Hypotenuse of the right triangle

The perimeter of a right triangle is calculated with the formula: P = a + b + c units.

How Do You Find the Perimeter of a Triangle With Coordinates?

If the coordinates of a triangle are given, then the length of all its sides can be calculated using the distance formula. Once these lengths are obtained, we can simply add them to find the perimeter of the given triangle.

How Do You Find the Perimeter of a Triangle With Two Equal Sides?

To calculate the perimeter of a triangle with two equal sides, we find the sum of lengths of all the sides. This type of triangle is called an isosceles triangle. The formula to calculate the perimeter of an isosceles triangle is 2a + b, where ‘a’ is the length of one of the equal sides, and ‘b’ is the length of the third side.

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:

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.

Example 1: Find the perimeter of the given triangle.

Here, sides of the triangle are AB = 3 cm, BC = 4 cm and AC = 3 cm

Perimeter of triangle = sum of all three sides

Therefore, the perimeter of the given triangle = 3 cm + 4 cm + 3 cm = 10 cm

Perimeter of an isosceles triangle:

Perimeter of an isosceles triangle = a + b + c

Here sides AB and AC are equal length so we can write the above formula as given below.

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

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

Perimeter of an equilateral triangle:

Perimeter of an equilateral triangle = a + b + c

As we know sides of an equilateral are equal in length.

So, the perimeter of an equilateral triangle = a + a + a= 3a

This formula can also be written as:

Perimeter of an equilateral triangle= 3 ✕ side

Using perimeter of a triangle in our everyday life

Following are the examples where we use the perimeter of a triangle in our daily life:

• The boundary of a triangular
• The boundary of a triangular hut

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