Introduction :

Three intersecting lines formed a closed figure is called a triangle. (‘Tri’ means ‘three’). As a triangle has three sides, it has three angles and three vertices. If the triangle having equal length in its all three sides then it’s called as equilateral triangle .The angles of an equilateral triangle are 60° each. "Equi" means that something same or equal; "lateral" refers to sides. Equilateral triangles are known as regular polygons and also equiangular in traditional Euclidean geometry.

**Study the properties of equilateral triangles:**

Let the lengths of the sides of the equilateral triangle are ‘a’, we can decide that:

The area of an equilateral triangle is A="a2(" v 3)/4

The perimeter of an equilateral triangle is P="3a

The radius of the circumscribed circle is

R=a (v 3)/3

The radius of the inscribed circle is r="a(v" 3)/6

The geometric center of the given triangle is the center of the circumscribed and inscribed circles

The altitude or height of the equilateral triangle is h="a" (v 3)/2.

Equilateral triangle is more symmetric.

Equilateral triangle has three lines of reflection.

Its rotational symmetry is order of 3.

Equilateral triangle belongs to the symmetry group of dihedral group of order 6 D3.**Study the Example problems on Equilateral triangle:**

Study equilateral triangle Ex1:

Find the area of the equilateral triangle of length 12cm.

Study equilateral triangle sol:

Let length = a="12" cm

The formula to find area of an equilateral triangle is A="(" v3/4) a2

A = ( v3/4) * 122

= 0.433 * 144

A = 62.352 cm2

Study equilateral triangle Ex 2:

The equilateral triangle having the length 6cm. Find its altitude.

Study equilateral triangle sol:

Let the length = a = 6cm.

Formula to find the altitude:

H= a (v 3/2)

= 6 (v 3/2)

= 6 * 0.866

Altitude = 5.196cm.

Study equilateral triangle Ex 3:

Find the perimeter of an equilateral triangle whole length is 13 cm.

Study equilateral triangle sol:

Let length = a="13" cm

Perimeter = P="3a

= 3 * 13

= 39 cm.

Study equilateral triangle Ex 4:

An equilateral triangle has one side that measures 25 in. What is the size of the angle opposite that side?

Study equilateral triangle sol:

Since it is an equilateral triangle each angles measures 60º. The measure of the angle does not depending upon on the length of the side.