Perimeter of Isosceles Triangle
A triangle is considered to be an isosceles triangle if it has two equal sides and two equal angles. The perimeter of an isosceles triangle is the total length of its boundary or the sum of all its sides. In this article, we will learn about the perimeter of an isosceles triangle using a few solved examples.
Perimeter of Isosceles Triangle
The perimeter of an isosceles triangle is the sum of all the three sides. Since an isosceles triangle has 2 equal sides, the perimeter is twice the equal sides plus the different side. It is measured in units such as inches (in), yards (yd), millimeters (mm), centimeters (cm), and meters (m). Let us understand the formula to find the perimeter in the next section.
Formula for the Perimeter of Isosceles Triangle
The perimeter of an isosceles triangle can be found by adding the length of all its three sides. Since an isosceles triangle has two equal sides, its perimeter can be calculated if the base and the equal sides are known. The formula used to find the perimeter of an isosceles triangle is: Perimeter of isosceles triangle (P) = 2a + b
where, a = the length of the two equal sides; b = the length of the base (unequal side)
Derivation of the formula: Observe the figure given below which will help us to derive the formula for the perimeter of an isosceles triangle. Let us consider an isosceles triangle ABC in which side AB = AC. Now, if the equal sides are named as 'a' and the unequal side is named as 'b', the sum of all its sides will be: AB + AC + BC, or, a + a + b. Therefore, the perimeter of an isosceles triangle is : 2a + b
Perimeter of Isosceles Right Triangle
The perimeter of an isosceles right angled triangle can be found by adding the length of all its three sides. Since it is a right angled triangle, one of its sides is the hypotenuse and the other two sides are equal. If the length of the hypotenuse is 'h' units and the lengths of the other two sides are 'l', then the perimeter of an isosceles right triangle would be: Perimeter an isosceles right triangle = h + l + l. Observe the following figure to understand the dimensions and the formula of an isosceles right triangle.
As given in the figure, the perimeter of an isosceles right triangle is P = h + 2l. On applying the Pythagoras theorem, we get h = √(l^{2}+ l^{2}) = √2 × l. This means h = √2 × l, which can also be written as: l = h/√2. These values can be substituted with the other if one of them is not known.Now, if the length (l) is given, then the perimeter of an isosceles right triangle will be (P) = 2l + (√2)l = (2 + √2)l
Similarly, if the hypotenuse (h) is given, then the perimeter of an isosceles right triangle will be (P) = h + 2(h/√2) = h + √2h
It should be noted that the two congruent angles in the isosceles right triangle measure 45° each.Related Topics
Check out these interesting articles to learn more about the isosceles triangle and its related topics:
Important Notes
Here is a list of a few points that should be remembered while studying the perimeter of an isosceles triangle:
 A triangle is considered to be an isosceles triangle if it has two equal sides.
 Since an isosceles triangle has two equal sides, its perimeter can be calculated if the base and the equal sides are known.
 The formula to calculate the perimeter of an isosceles triangle is P = 2a + b where 'a' is the length of the two equal sides and 'b' is the base of the triangle.
 The perimeter of an isosceles right triangle can be calculated with the help of the formula: P = h + 2l, where 'h' is the length of the hypotenuse and 'l' is the length of the adjacent sides.
Solved Examples

Example 1: What is the perimeter of an isosceles triangle when the length of one of the equal sides = 10 cm and base = 6 cm?
Solution:
Given: length of one equal side (a) = 10 cm; base (b) = 6 cm
The perimeter of an isosceles triangle = 2a + b, where a = 10, b = 6. Substituting the values in the formula:
= 2 (10) + 6
= 20 + 6
= 26 cmAnswer: The perimeter of the given isosceles triangle is 26 cm.

Example 2: The perimeter of an isosceles triangle is 12 units. Find out the third side of the triangle if the equal sides measure 3 units each.
Solution: Let the third side of the isosceles triangle be 'b'
Given, Perimeter = 12 units, a = 3 units, b = ?
Perimeter of an isosceles triangle = 2a + b
Substituting the given values in the above formula,
12 = 2 (3) + b
12 = 6 + b12  6 = b
Therefore, b = 6 units
Answer: The third side of the triangle measures 6 units.
FAQs on the Perimeter of the Isosceles Triangle
What is the Formula to Calculate the Perimeter of an Isosceles Triangle?
The formula to calculate the perimeter of an isosceles triangle is: Perimeter = 2a + b where 'a' represents the length of the two equal sides and 'b' is the base.
What is the Formula to Calculate the Perimeter of an Isosceles Right Triangle?
The perimeter of an isosceles right triangle is calculated with the help of the formula: P = h + 2l, where 'h' is the length of the hypotenuse and 'l' is the length of the adjacent sides. Since it is a right angled triangle, we can apply the pythagoras theorem and calculate the perimeter using only the length or the hypotenuse, whichever is given. The formula using only the length is: P = (2 + √2)l. Similarly, the formula using only the hypotenuse is: P = h + √2h.
What is the Formula to Calculate the Perimeter of a Triangle?
The perimeter of a triangle can be found by calculating the sum of all its sides. So, the formula for calculating the perimeter of a triangle with side lengths a, b, and c, would be: Perimeter = a + b + c.
How to Find the Perimeter of an Isosceles Triangle If the Area Is Given?
The perimeter of an isosceles triangle can be calculated if the area and height are known. In this case, we can simply follow the steps given below,
Step 1: Using the area, "A", and height, "h", base, "b" of the isosceles triangle can be calculated as, b = 2A/h
Step 2: We can use this value of base and height while applying Pythagoras theorem to find the measure of equal sides, "a", a = √[h^{2} + (A/h)^{2}]
Step 3: Finally, find the sum of all three sides to find the perimeter of isosceles triangle as, P = 2a + b = 2√[h^{2} + (A/h)^{2}] + 2A/h
How to Find the Perimeter of an Isosceles Triangle With Base and Height?
We can find the perimeter of an isosceles triangle using base and height by following the steps given below:
 Step 1: Using the base, "b", and height, "h", of an isosceles triangle, we can find out the measurement of the remaining equal sides, "a". In this case, we will apply the Pythagorean theorem and find the missing side, given as, a = √[h^{2} + (b/2)^{2}].
 Step 2: Finally, find the sum of all the sides to calculate the perimeter of the given isosceles triangle,
P = 2b + b = b + 2√[h^{2} + (b/2)^{2}] = b + √[4h^{2} + b^{2}]
How to Find the Perimeter of an Isosceles Triangle on a Coordinate Plane?
Given the coordinates of the vertices of an isosceles triangle, we can calculate the perimeter by finding the length of all sides and then further adding them. To calculate the lengths using coordinates, we apply the distance formula and finally find their sum.