Competitive exams in the United States typically have questions on geometry to assess a student's knowledge of simple shapes and their attributes. One such fundamental idea is the perimeter of a triangle. The term "perimeter" is derived from the Greek terms peri (around) and metron (measure), which mean the total distance around a figure. For a triangle, its perimeter is the sum of the lengths of its three sides. Knowing how to compute the perimeter is not just significant in performing well on standardized exams such as the SAT and ACT but also in solving actual problems, such as approximating the fencing for a garden or finding the border of a sporting field. This article discusses the perimeter of a triangle in-depth, including formulas for various kinds of triangles, how to find the perimeter step by step, and numerous examples solved to further your knowledge.

Perimeter of Triangle

The perimeter of any two-dimensional figure depends on the area around it. Any closed shape’s perimeter can be calculated by adding the lengths of its sides. The perimeter of a polygon is given by the sum of all its sides. We know that the length of each side of a regular polygon is the same.

The perimeter of a regular polygon is = the sum of all its sides = the number of sides ✕ length of one side.

A triangle is a polygon with three sides. Thus, the perimeter of a triangle is actually the total length of all its three sides. The perimeter of a triangle is the total distance covered along the boundary of any triangle.

Any plane figure’s perimeter can be thought of as the sum of its boundaries. Since the perimeter is a length measure, it is expressed in m, cm, km, etc. A triangle is a polygon with three sides, and there are several ways to calculate its perimeter.

We often use the concept of the perimeter in real life. For example, when putting up Diwali lights around the house or when we want to put a fence around the backyard, we find its perimeter to know the length of wire we will need.

Perimeter of Triangle Formula

The perimeter of the triangle is P= a + b + c, where a, b, and c are the sides of the triangles. Any two-dimensional figure’s perimeter is defined as the distance around it. By combining the lengths of each of the sides, we can compute the perimeter of any closed shape.

Perimeter = Sum of the three sides in the case of a triangle

The perimeter of a triangle can be calculated by adding the lengths of the three sides of the triangle. But there are different methods as well to calculate the perimeter of different types of triangles.

Let’s see this with the help of an example:

Example: Calculate the perimeter of the given triangle.

We know that the perimeter of a triangle is given by

Perimeter = A + B + C,

Where A, B, C = length of three sides.

Therefore,

For the given triangle,

Perimeter = 5 cm + 4 cm + 3 cm = 12 cm

Perimeter of Equilateral Triangle

In geometry, an equilateral triangle is a triangle in which all three sides have the same length. Because all the sides of an equilateral triangle are equal to each other, therefore the perimeter of an equilateral triangle can also be calculated by the formula:

Perimeter, P= 3L

Where L is the length of the sides of the triangle.

We can easily derive this formula from the main formula of the perimeter of the triangle.

As seen above the perimeter of any triangle is the sum of all sides. However, in the case of an equilateral triangle, all sides are of the same length. Thus, A = B = C = L.

Hence, Perimeter = A + B + C = L + L + L = 3L

Perimeter of Isosceles Triangle

In geometry, an isosceles triangle is a triangle that has at least two sides of equal length. Because two sides of an Isosceles triangle are equal and one side is of a different length than the other two, therefore the perimeter of an isosceles triangle can also be calculated by the formula:

Perimeter, P= (2 × A) + B

Where A is the length of two equal sides and B is the length of the third side.

This too can be derived from the main formula of the perimeter of the triangle.

As seen above the perimeter of any triangle is the sum of all sides. However, in the case of an isosceles triangle, two sides are of the same length. Thus, A = C.

Hence, Perimeter = A + B + C = A + B + A = P= (2 × A) + B.

Perimeter of Right Angle Triangle

The simplest formula for finding the area of a right angled triangle is by adding all the sides together. However, in some cases, one or two sides might be unknown. Thus, we have to apply some theorems or use some formulas to find the perimeter. Which theorem we apply depends on the data available to us. If only one side is unknown, we can simply apply the Pythagoras Theorem. However, if two sides are unknown we have to use the law of sines.

Perimeter of Right Angle Triangle using Pythagoras’ Theorem

Pythagoras’ theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

\(c^2 = a^2 + b^2\)

Where c is the hypotenuse and a and b are the other two sides.

Here’s an example that will help you understand better.

Example: Calculate the perimeter of the given triangle.

In the above image, a = 4; b= 3; c is unknown. We can find this by the Pythagoras theorem.

\(c^2 = 4^2 + 3^2\)

\(c = \sqrt{25}\)

\(c = 5\)

The Law of Sines is applicable to all sorts of triangles and is explained briefly in the next few sections.

Perimeter of Isosceles Right Triangle

A triangle in which one angle measures 90°, and the other two angles measure 45° each is an isosceles right triangle. In an isosceles right triangle (figure below), ∠A and ∠C measure 45° each, and ∠B measures 90°. The sides AB and AC are equal. This type of triangle is also known as a 45-90-45 triangle

AC, the side opposite of ∠B is the hypotenuse. And AB or AC can be taken as height or base

The Sum of all sides of a triangle is the perimeter of that triangle.

In the right isosceles triangle, since two sides (Base BC and Height AB) are the same and taken as ‘B’ each. And the hypotenuse is taken as ‘H’.

The perimeter= AB + BC + AC

= B+B+H

=2B+H

Perimeter of Scalene Triangle

A scalene triangle is a triangle in which all three sides are of different lengths, and all three angles are of different measures. Because the sides of a scalene triangle are all of different lengths, therefore it can only be calculated by the conventional formula:

Perimeter of Scalene Triangle, P= A + B + C

where A, B, and C are the lengths of the sides of a triangle.

Perimeter of Triangle from Vertices

A point where any two sides of a triangle meet are called a vertex of a triangle. We can find the perimeter of a triangle by knowing the coordinates of its vertices. Consider a triangle ABC where \(A\equiv(x_1, y_1), B\equiv(x_2, y_2)\) and \(C\equiv(x_3, y_3)\) are the vertices of the triangle. The side lengths are the lengths of the line segments AB, BC, and CA. These may be found using the endpoints of the segments with the distance formula:

You know that the distance AB between two points in a plane with Cartesian coordinates \(A\equiv(x_1, y_1)\) and \(B\equiv(x_2, y_2)\) is given by the following formula:

\(AB=\sqrt{(x_2−x_1)^2+(y_2−y_1)^2}\)

The distance formula is really just the Pythagorean Theorem in disguise.

Similarly, we can calculate the distance between points B and C and A and C.

\(BC=\sqrt{(x_3−x_2)^2+(y_3−y_2)^2}\) and

\(AC=\sqrt{(x_1−x_3)^2+(y_1−y_3)^2}\)

Now that we know the length of the sides we can easily find out the perimeter by adding the distances together.

Thus, Perimeter = AB + BC + AC.

Perimeter of Triangle When One Side is Unknown (Law of Cosines)

The Law of Cosines can be used when anyone’s side is unknown and we know the measure of the opposite angle. This law can be applied to any triangle in general.

\(c = \sqrt{a^2+b^2-2abcosy}\)

In the above equation, the unknown side is c, a and b are known, and y is the angle opposite to the unknown side. If c is the hypotenuse, then y will be 90o. As cos(90) is zero, hence the equation reduces to \(\sqrt{a^2 + b^2}\). This proves that this law is consistent with Pythagoras’ theorem.

Perimeter of Triangle When Two Sides are Unknown (Law of Sines)

This law can be used when at least two angles are known along with two sides. This law can also be used in combination with other laws and formulas to find the perimeter of any triangle. This law also gives the relationship between the sine of angles and sides of a triangle.

\(\frac{sinx}{A} = \frac{siny}{B} = \frac{sinz}{C}\)

Here A, B and C are sides opposite to angles x, y and z respectively.

Semi-Perimeter of Triangle

In geometry, the semi-perimeter of a polygon is half its perimeter. We know that the perimeter of a shape is the distance around it, but the semi-perimeter is half the distance around it. The semi-perimeter of a given polygon can be computed by dividing its circumference by two for each given polygon. Although it has such a simple derivation from the perimeter, the semi-perimeter appears frequently enough in formulas for triangles and other figures that it is given a separate name. When the semi-perimeter occurs as part of a formula, it is typically denoted by the letter s.

Area of Triangle from its Perimeter (Heron’s Formula)

Most formulas use the height of a triangle to calculate the area. When the height of the triangle is not known, then we use Heron’s formula to calculate the area of the triangle. Heron’s formula gives the area of a triangle by requiring no arbitrary choice of the side as base or vertex as origin, contrary to other formulas that calculate the area of a triangle. The area can be calculated using all three sides where none of the sides is equal.

\(\text {Area} =\sqrt{s(s-a)(s-b)(s-c)}\)

where, \(a, b\) and \(c\) are the lengths of sides of the triangle and \(s\) is the semi-perimeter of the triangle, given by \(s=\frac{a+b+c}{2}\).

We have already seen what is meant by the semi-perimeter of a triangle. Thus, Heron’s formula can give us the semi-perimeter of a triangle if the area and sides are known. From this semi-perimeter, one can easily calculate the perimeter of the triangle.

Steps to Find Perimeter of Triangle

The steps to find the perimeter of a triangle depend on the available data. Remember the following few points while finding the perimeter of a triangle.

• Simple method: Add all sides
• One side Unknown: Use the Pythagoras theorem to calculate the unknown side in the case of the right angled triangle. Use the law of cosines to find the unknown side in the case of any other triangle.
• Use the law of sine if two sides of the triangle are unknown.
• Use the formula for the area if the area of the triangle is known.

The steps are as listed below:

Step 1: Calculate/ Measure/ Note all the lengths of the three sides.

Step 2: Add them together.

Step 3: Give the proper units at the end.

Perimeter vs Area of Triangle

The perimeter is the sum of all the side lengths of a shape. The area is the amount of two-dimensional space a shape occupies. To better understand these two concepts and their differences, let’s look at an example:

Consider a triangular piece of land of 10 m sides. The owner wants to cover the land with green grass and put a fence around the land. He hires a landscaper to do so. The landscaper needs to order the supplies. How will he calculate the number of grass mats and wooden fences required? The answer is simple: By calculating the area and perimeter of the triangle. In this case, the number of grass mats required will be given by the area of the triangle as it will give us the amount of space covered by the land. On the other hand, the perimeter will tell us about the length of the fence.

One must also remember that ‘A triangle with a fixed perimeter can have many different areas’.

Solved Examples on Perimeter of Triangle

Now that we have learned about the perimeter of the triangle, let’s see some solved examples of it.

Example 1:  Given the perimeter of an equilateral triangle is 24cm, find the length of its three sides.

Solution: The perimeter of an equilateral triangle is equal to three times the length of a side since each of the triangle’s three sides is the same length.

Let the length of any side be equal to ‘a’ units. So perimeter is equal to ‘3a’ units.

So, we can write,

3a = 24

a = 8cm

Thus, the length of each side is equal to 8cm.

Example 2: If the perimeter of a triangle is 40 cm and the length of two of its sides is 8 cm each, what will be the length of the third side of the triangle?

Solution: We are given the value of two sides of the triangle that are equal; hence, it is an isosceles triangle.

The perimeter of an isosceles triangle = 2a + b

\(48 = (2\times 8) + b\)

\(b = \frac{48}{16}\)

b = 3 cm

Example 3: Consider a right angle triangle ABC where the side AC is the hypotenuse. If the measurements of the sides AB and BC are 3 cm and 4 cm respectively, what will be the perimeter of the triangle?

Solution: We need the values of all three sides to calculate the perimeter of the right triangle. As this is a right triangle, we can calculate the length of side AC using the Pythagorean theorem.

\(AC^{2} = AB^{2}+BC^{2}\)

\(AC = \sqrt{(AB^{2}+BC^{2})}\)

\(AC = \sqrt{(3^{2}+4^{2})}\)

\(AC = \sqrt{9+16}\)

\(AC = \sqrt{25}\)

AC = 5 cm

Perimeter = AB + BC + AC

Perimeter= 3 + 4 + 5

Perimeter = 12 cm

In conclusion, understanding the perimeter of a triangle is an essential skill for high school students preparing for U.S. competitive exams like the SAT, ACT, and other standardized tests that include geometry concepts. The term perimeter comes from the Greek words peri (around) and metron (measure), describing the total distance around a two-dimensional shape. In the case of a triangle, the perimeter is simply the sum of all its sides. Knowing how to calculate a triangle’s perimeter is useful not only for acing exam questions but also for real-life situations—like figuring out how much fencing is needed for a backyard or how much string is required to mark the boundary of a soccer field. By mastering these formulas and applying them correctly, students can boost their confidence and problem-solving abilities, helping them perform better in their exams.