# Perimeter of Isosceles Triangle – Definition, Examples, Facts, FAQs

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## What Is the Perimeter of an Isosceles Triangle?

The perimeter of an isosceles triangle is the sum of all its sides. It is given by $2x + y$, where x is the length of two equal sides and y is the length of the remaining side.

The perimeter is defined as the total length of the boundary of a shape. It is the sum of all the sides of the shape. A triangle is a polygon with three sides. So, what is the formula for perimeter of a triangle? It is simply the sum of all three sides of the triangle.

An isosceles triangle is a type of triangle that has two sides of the same length. Also, the angles opposite to the equal sides (known as the base angles) are congruent.

In the given $\Delta ABC$, AB and AC are two equal sides. Angles B and C are congruent.

## Perimeter of an Isosceles Triangle Formula

The formula for the perimeter of an isosceles triangle is given by

Perimeter $= 2x + y$

where, x is the length of the equal sides and y is the unequal side of the isosceles triangle.

Perimeter is measured in linear units such as inches, yards, etc., same as the length of the sides of the triangle.

## Derivation of the Formula for Perimeter of an Isosceles Triangle

We know that the perimeter of a triangle is the sum of all its sides. Thus, if the side-lengths of a triangle are given by x, y, and z, then we can find the perimeter as

Perimeter of a triangle $= (x + y + z)$ units

Now, take a look at the isosceles triangle shown below where two sides are equal.

Thus, substitute $x = z$

Perimeter of an isosceles triangle $= x + y + x = (2x + y)$ units

## How to Find the Perimeter of an Isosceles Triangle

Step 1: Note down the lengths of sides. Let the “x” be the length of equal sides, and “y” be the length of the unequal side. Make sure that all the sides have the same unit.

Step 2: Substitute the values in the formula.

Perimeter of an isosceles triangle $= (2x + y)$ units.

Step 3: Assign the appropriate unit, which will be the same as the length of the sides.

## Perimeter of Isosceles Right Triangle

Isosceles right triangle, as the name itself suggests, is a triangle with one right angle and two equal sides.

In the given isosceles right triangle PQR, the sides PQ and QR are equal. These sides represent the legs (base and altitude) of the right triangle. The angle Q is a right angle.

Perimeter of an isosceles right triangle $= 2l + h$

If we don’t know the length of the hypotenuse, we can find it using Pythagoras’ theorem.

By Pythagoras’ theorem,

$Hypotenuse^{2} = Base^{2} + Perpendicular^{2}$

Thus, $h^{2} = l^{2} + l^{2}$

$h^{2} = 2l^{2}$

$h = \sqrt{2\;l}$

Calculating the Perimeter of an Isosceles Right Triangle if Hypotenuse Is Given