The isosceles triangle is a type of triangle, which has two sides with the same length. An isosceles triangle two angles will also be the same in front of the equal sides. If all three sides are equal in length then it is known as an equilateral triangle. Therefore we may conclude that all equilateral triangles also have all the properties of an isosceles triangle. In this article, we will discuss the isosceles triangle and various isosceles triangle formula. Let us begin learning!

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**Isosceles Triangle Formula**

**What is the Isosceles Triangle?**

An isosceles triangle is a triangle with two sides of equal length and two equal internal angles adjacent to each equal sides. It is unlike the equilateral triangle because there we can use any vertex to find out the altitude of the triangle. Thus in an isosceles triangle to find altitude we have to draw a perpendicular from the vertex which is common to the equal sides.

Also, in an isosceles triangle, two equal sides will join at the same angle to the base i.e. the third side. These special properties of the isosceles triangle will help us to calculate its area as well as its altitude with the help of a few pieces of information and formula. Here, the student will learn the methods to find out the area, altitude, and perimeter of an isosceles triangle.

**Some Properties**

- The unequal side of an isosceles triangle is normally referred to as the base of the triangle.
- The base angles of the isosceles triangle are always equal.
- If the third angle is the right angle, it is called a right isosceles triangle.
- The altitude of a triangle is a perpendicular distance from the base to the topmost

**The Formula for Isosceles Triangle**

**The perimeter of an Isosceles Triangle:**

P = 2Ã— a + b

Where,

P | Perimeter |

a | The measure of the equal sides |

b | The base of the triangle |

**Area of an Isosceles Triangle:**

A = \(\frac{1}{2} \times b \times h \)

Where,

A | Area |

a | The measure of the equal sides |

b | The base of the triangle |

**The altitude of an Isosceles Triangle:**

h = \(\sqrt(a^2 – \frac{b^2}{4}\)

h | Altitude or height of the triangle |

a | The measure of the equal sides |

b | The base of the triangle |

**Solved Examples**

Example-1: Calculate Find the area, altitude, and perimeter of an isosceles triangle. Its two equal sides are of length 6 cm and the third side is 8 cm.

Solution:

As given in the problem we have,

a = 6 cm

b = 8 cm

First, we will compute Perimeter of the isosceles triangle using formula,

P = 2Ã— a + b

P = 2Ã— 6 + 8

= 20 cm

**Therefore perimeter will be 20 cm.**

Now, we will compute the Altitude of the isosceles triangle as follows,

h = \(\sqrt(a^2 – \frac{b^2}{4})\)

h = \(\sqrt(6^2 – \frac{8^2}{4})\)

So,

h = \(\sqrt (36-16)\)

h = \(2\sqrt5\; cm \)

Thus altitude of the triangle will be \(2\sqrt5 \; cm. \)

Finally, we will compute the Area of the isosceles triangle as follows,

A = \(\frac{1}{2} \times b \times h \)

= \(\frac{1}{2} \times 8 \times 2\sqrt5\)

= \(8 \sqrt5 \; square\; cm \)

Therefore the triangle will have area of \(8 \sqrt5 \; square\; cm .\)

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