The perimeter of any triangle is the measurement of the entire distance around it. The perimeter of a triangle provides information to mathematicians on the type and dimension of shape they are working. The perimeter of the triangle typically provides information on distance, unlike the Area that discusses the deeper aspects of the shape.
There are different ways to calculate the perimeter of the triangle, and this article will comprehensively consider all of them to teach readers and students how to calculate it.
The Triangle
The triangle is a three-sided polygon that can be presented in different forms. There’s the basic triangle, right-angle triangles, obtuse triangles, etc. Diagrammatically, triangles are expressed as
There are other types of triangles, and they all share a common characteristic of having three sides. For the basic triangle, both lengths labeled as ‘a’ and ‘c’ are the same, while the ‘b’ is the base. The right-angled triangle, the part abled as ‘a’ is the length, b is always the base, while the part labeled as ‘c’ is known as the Hypotenuse.
The Perimeter of a Triangle Formula
The perimeter of a triangle is the sum of the side lengths of the triangle. The basic formula for the perimeter of the triangle if all side lengths are given is;
P = a + b + c
Where each variable represents its respective sides in the triangle diagram above
The above formula calculates the perimeter of triangles when the side lengths are available. There may be instances where a side length may be missing. You could use two formulas to find the missing lengths depending on the type of the triangle.
One of the methods to use is the Pythagoras Theorem;
The square of the Hypotenuse = The Square of the Opposite + The Square of the Adjacent
i.e.
c2 = a2 + b2
This formula is only applicable for finding the side lengths of the right-angle triangle.
The second method is the Rule of Cosines
c2 = a2 + b2 – 2ab Cos(d)
where ‘c’ is the Hypotenuse, ‘a’ is the length, ‘b’ is the base, and ‘d’ is the degree of the triangle
This formula is only applicable for the obtuse triangle or any triangle with a degree.
It is important to know that the perimeter of the triangle cannot be determined if there are no side lengths to work with. Typically, there should be at least two sides available to work with.
How to Calculate the Perimeter of a Triangle?
While there is a basic formula to calculate the perimeter of the triangle, certain conditions could affect the steps.
Method 1: Using the Basic Formula
Step 1: Ensure that all the sides lengths are available
Step 2: Use the perimeter of a Triangle formula given as P = a + b + c
Method 2: Using the Pythagoras Theorem for one Missing side lengths
Step 1: Ensure the Triangle is a Right angle
Step 2: Determine the missing side length by using the Pythagoras Theorem to find it
Step 3: Use the perimeter of a Triangle formula given as P = a + b + c
Method 3: Using the Cosine Rule for one Missing Side Lengths
Step 1: Ensure that the degree of the triangle is available
Step 2: Use the given formula for the law of cosines to c2 = a2 + b2 – 2ab Cos(d) to find the missing length.
Step 3: Find the Perimeter of the triangle using the Formula; P = a + b + c
Calculating the Perimeter of a Triangle with Examples
Example 1
Find the perimeter of the triangle below
Solution
Step 1: Ensure that all the sides lengths are available
From the above triangle, all the side lengths are available, so the simple perimeter type of calculation can be used directly.
Step 2: Use the perimeter of a Triangle formula given as P = a + b + c
P = a + b + c
Where;
a = 11 cm,
b = 10 cm,
c = 11 cm
P = 11 cm + 10 cm + 11 cm
P = 32 cm
Therefore, the perimeter of the triangle is 32 cm.
Example 2
Consider the right angle triangle below and find its perimeter
Solution
Considering the above triangle, it is obvious that it is a right-angle triangle with a missing base. This means that the Pythagoras theorem can be used to find the base. We will use the second method outlined in this article.
Step 1: Ensure the Triangle is a Right angle
This has been determined
Step 2: Determine the missing side length by using the Pythagoras Theorem to find it
The Pythagoras theorem is given as;
c2 = a2 + b2
where a = 9 cm, b = ?, c = 15 cm
152 = 92 + b2
225 = 81 + b2
225 – 81 = b2
144 = b2
b = 12
The missing length has been determined with the Pythagoras theorem. We can now proceed to find the triangle’s perimeter.
Step 3: Use the perimeter of a Triangle formula given as P = a + b + c
P = 9 + 12 + 15
P = 36
Therefore, the perimeter of the right angle triangle is 36 cm
Example 3
Find the perimeter of the triangle
Solution
From the triangle, it is obvious that one of the side lengths is missing. So we will either use the Pythagoras theorem or the rule of cosines. As can be seen from the shape of a triangle., it is neither a basic triangle nor a right angle triangle. So it is clearly an obtuse triangle. Also, there is a degree, and this means that rule of cosines is the best formula to go for.
Step 1: Ensure that the degree of the triangle is available
The degree of the formula = 970,
a = 11 cm
b = 13 cm
c = ?
Step 2: Use the given formula for the law of cosines to c2 = a2 + b2 – 2ab Cos(d) to find the missing length.
c2 = a2 + b2 – 2ab Cos (d)
c2 = 112 + 132 – 2 x 11 x 13 x Cos (97)
c2 = 121 + 169– 2 x 11 x 13 x Cos (97)
c2 = 121 + 169– 286 x Cos (97)
now Cos 97 = – 0.92514; therefore
c2 = 121 + 169– 286(- 0.92514)
c2 = 121 + 169+ 264.5922
c2 = 554.5922
c = 23.5498
Therefore, the missing length is c = 23.5498
Step 3: Find the Perimeter of the triangle using the Formula; P = a + b + c
P = 11 cm + 13 cm + 23.5498 cm
P = 47.545 cm
Example 4
Consider an obtuse angle with a perimeter given as 62 cm if the height of the angle is 26 cm and the length 22 cm. Find the degree of the triangle.
Solution
Before attempting to start any calculation, it is important to understand some major points.
The aim of this problem is to find the degree of the obtuse triangle. We can use the cosine rule to find the degree as it is the only formula with a degree in it and can be used for obtuse triangles.
The formula for the cosine rule is given as
C2 =a2 + b2 – 2abCos(d)
Where c = 26 as hypotenuse is the same as height
a = 22 cm
b = ? as it is not made available in the information given
c = 26 cm
d = ?
Inputting the above into the cosine rule formula gives
262 =222 + b2 – 2 x 22 x b x Cos(d)
676 = 484 – 44b x Cos (d) ………… (1)
From the above solving, it is obvious that further computation is not possible due to the base of the triangle missing.
Now recall that the perimeter of the triangle was given in the info provided. So were the length and Hypotenuse. Now since all three values are made know substituting them into the perimeter formula can help us find the base ‘b’
The perimeter of the triangle is given as;
P = a + b+ c
As such;
52 = 22 + b + 26
62 = 48 + b
62 – 48 = b
14 = b.
Since we have been able to find the base using the perimeter of the triangle formula, we will substitute it into the Cosine rule.
676 = 484 – 44b x Cos (d) …….. (1)
676 = 484 – 44(14) x Cos (d)
676 = 484 – 616 x Cos (d)
676 – 484 = – 616cos (d)
192 = – 616cos (d)
– 192/616 = cos (d)
-0.312 = Cos (d)
Cos-1(-0.312) = d
d = 1.8890
Conclusion
The perimeter of the triangle is very easy to compute. The major factor that students only need to consider is the type of triangle and the appropriate method to use. This article considered three possible types of triangles and the respective steps and methods needed to calculate them. Four examples were also considered in this piece to explain how the methods work.