 # Isosceles Triangle Formula

## What is a Triangle?

A triangle is a 2-dimensional closed figure that has three sides and angles. It is the basic or the purest form of Polygon. We can draw a triangle using any three dots in such a way that the line segments will connect each other end to end. Those three line segments are the sides of the triangle, the point where the two lines intersect is known as the vertex, and the space between them is what we call an angle. We should also know that the sum of all the interior angles of a triangle is always 180 degrees.

### Types of Triangle

Triangles are classified into two categories based on their side and angle.

Triangles based on their sides can be:

1. An equilateral triangle

2. Isosceles triangle

3. Scalene triangle.

Triangles based on their angles can be:

1. Acute angled triangle

2. Right-angled triangle

3. Obtuse angled triangle

### What is the Isosceles Triangle?

If two out of three sides of a triangle have equal length, then the triangle will be called an isosceles triangle. Therefore, the two opposite sides in an isosceles triangle are equal. As we already know that the sum of all the angles of a triangle is always 180,so if two of the sides of a right-angled triangle are known to us, we can find the third side of the triangle. Here, given below, is an example of a right-angled triangle.

What are the Properties of an Isosceles Triangle?

Now that we know what a triangle and an isosceles triangle is, it’s best if we move on the question, what are the properties of an isosceles triangle. So here are the properties of a right-angled triangle.

1. In an isosceles triangle, the two sides are congruent to each other.

1. The side of the triangle that is unequal is called the base of the triangle.

2. In an isosceles triangle, the height that is drawn from the apex divides the base of the triangle into two equal parts and the apex angle into two equal angles.

3. The formula to find the area of isosceles triangle or any other triangle is: ½ × base × height.

4. To find the perimeter of the triangle we just have to add up all the sides of the triangle.

5. The unequal angle or the base of the triangle is either an acute or obtuse angle.

6. The sides of the triangle form the chords of the circumcircle.

### Formulas For an Isosceles Triangle

There are two formulas for an isosceles triangle, one is to find the area of the triangle, and the other is to find the perimeter of an isosceles triangle.

### Area of Isosceles Triangle

The formula to calculate the area of isosceles triangle is:

= $\frac{b}{2} \sqrt{a^{2} - \frac{b^{2}}{4}}$

Since in an isosceles triangle, we know that the two sides of it are equal and the base of the triangle is the unequal one. Therefore, we can use the following formula to find the area of an isosceles triangle.

### Perimeter of Isosceles Triangle

Just like finding the perimeter of any other figure is easy, the perimeter of isosceles triangle is also very easy. We have a straightforward and familiar formula to find the perimeter of isosceles triangle. To find the perimeter, we just have to add all the sides of the triangle, i.e., side 1 + side 2 + side 3. In the case of an isosceles triangle, we know the two sides of it are equal, so the formula for finding the perimeter an isosceles triangle can be modified as 2a + c if the sides are a, b, and c.

### Fun Facts

1. In Latin, the word isosceles is written as “ ‘īsoscelēs,’ and in Greek, the word isosceles is written as ‘ἰσοσκελής (isoskelḗs).’ In both Latin and Greek, the meaning of the word ‘isosceles’ means  “equal-legged”.

2. Even before the Greek mathematician could study the isosceles triangle, the  Babylonian and Egyptian already knew how to calculate the area od an isosceles triangle.

3. The buildings in the shape of an isosceles triangle are not only attractive but earthquake-resistant as well.

### Solved Examples

Example 1) Given below is a figure of an equilateral triangle(ABC) and an isosceles triangle (ABD) where DA=DB. What is the isosceles triangle’s unknown angle x?

Solution 1) Since we have an equilateral triangle (ABC)

Therefore, ∠CAB is equal to ∠ ABC which is equal to ∠ BCA which is equal to 60°

and ∠ DBA is equal to ∠ DAB ∠ (60°- x).

In ΔDAB we have,

∠DBA + ∠ DAB + ∠ ADB is equal to 180°

⇒ 2 (60° – x) + 88° = 180°

⇒ 2 (60° – x) = 92°

⇒ 60° – x = 46°

⇒ x = 14°

Example 2) PQR is a triangle where QR=PR and ∠P = 36°. So which side, according to you is the most significant side of the triangle?

Solution 2) Here, it is known is that if QR is equal to PR, then the angles that are opposite to them has to be equal as well, which means that  ∠P =  ∠Q =   36°.

Another key point that we already know is that the sum of all the angles in a triangle is equal to 180°. Therefore, ∠R = 180° –  36° – 36° = 180°.

Out of these three angles, ∠R is the largest which means that the side which is opposite to it that is PQ is the most significant side of the triangle.