An equilateral triangle is a triangle whose all three sides are having the same length. This is the only regular polygon with three sides. It appears in a variety of contexts, in both basic geometries as well as in many advanced topics such as complex number geometry and geometric inequalities. Let us start learning about the equilateral triangle formula.

## What is an Equilateral Triangle?

We can split the word equilateral into two words as ‘equi’ meaning equivalent and lateral meaning side. Therefore, an equilateral triangle is simply a triangle whose three sides are all equal. It is obvious that along with this triangle’s sides, all three angles are also equal. As we know that the sum of a triangle’s angles is always 180 degrees. Thus each angle in an equilateral triangle will be 60 degrees.

Hence, we can see that the equilateral triangle is the unique polygon for which by knowing only one side length one can determine the full structure of the polygon. In other words, the equilateral triangle is in company with the circle and the sphere whose full structures are known only by knowing the radius.

## Equilateral Triangle Formula

Formulas for the area, altitude, perimeter, and semi-perimeter of an equilateral triangle are as given:

Where,

a is the side of an equilateral triangle.

h is the altitude of an equilateral triangle.

(1) h = \(\frac{\sqrt{3}a}{2}\)

(2) A= \(\frac{\sqrt{3}a^2}{4}\)

(3) p = 3a

(4) s = \(\frac{3a}{2}\)

Where,

A | Area of the equilateral triangle |

p | The perimeter of an Equilateral Triangle |

s | Semi Perimeter of an Equilateral Triangle |

a | Length of one side |

h | Height of triangle |

**Derivation of the Equilateral Triangle Formula**

Let one side length of the equilateral triangle is “a” units. Hence the derivation of the equilateral triangle formula is as follows-

As we know that the area of Triangle is given by;

A = \(\frac{base\times height}{2}\)

Also, drawing a perpendicular from vertex to the base, will divide the triangle into two equal right-angled triangle. This triangle will have base length a/2 and hypotenuse length as a. thus base = a

Also length of perpendicular i.e. h will be,

\(h^2 = a^2 -(\frac{a}{2})^2\), using Pythagorean theorem.

i.e. \(h^2 = \frac{3a^2}{4}\)

i.e. \(h= \frac{\sqrt{3}a}{2}\)

Thus

**h = \(\frac{\sqrt{3}a}{2}\)**

hence Formula – 1 proved.

As we know that the area of Triangle is given by;

A = \(\frac{base\times height}{2}\)

i.e. A = \(\frac{a\times\frac{\sqrt{3}a}{2}}{2}\)

**i.e. A= \(\frac{\sqrt{3}a^2}{4}\)**

Hence Formula – 2 proved

Also perimeter = sum of all side lengths

So, p = a+a+a

**i.e. p =3a .**

Hence Formula – 3 proved.

Thus semi-perimeter, which will be half of the perimeter,

**s= \(\frac{3a}{2}\)**

Hence Formula – 4 proved.

## Solved Examples

Q.1: Find the area of an equilateral triangle with a side of length 7 cm?

Solution:

Given,

Side of the equilateral triangle i.e.

a = 7 cm

Also, area of an equilateral triangle,

A= \(\frac{\sqrt{3}a^2}{4}\)

i.e. A= \(\frac{\sqrt{3}7^2}{4}\)

i.e. A= \(49\times \frac{\sqrt{3}}{4} square cm\)

i.e. A = 21.21762 square cm

Thus area = 21.21762 square cm.

Q.2: Find the height of an equilateral triangle whose side is 28 cm?

Solution:

Given,

Side of the equilateral triangle,

i.e. a = 28 cm

We know, height of an equilateral triangle,

i.e. h= \(\frac{\sqrt{3}a}{2}\)

i.e. h= \(\frac{\sqrt{3}\times 28}{2}\)

i.e. h= \(14\sqrt{3} cm\)

Thus height = \(14\sqrt{3} cm\)