In this topic, we will discover about equilateral triangles and its area. The student will also learn the area of equilateral triangle formula. 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!

**Area of Equilateral Triangle Formula**

**What is the 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.

**Area of Equilateral Triangle Formula:**

The area is the size of a two-dimensional surface. The area of a plane surface is a measure of the amount of space covered by it. Calculating areas is a very important skill used by many people in their daily work. This are computation is highly dependent on the shape and size of the object. For the triangle, we have the formula to find out its area. For the general triangle, it is a little bit of complex calculation. But, finding the area of an equilateral triangle is comparatively easy.

Area of an equilateral triangle can be computed by the formula:

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

Where

A | Area of Equilateral triangle |

a | Side length |

**Derivation of the formula:**

Let one side length of the equilateral triangle is “a” units.

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 height = \(\frac{\sqrt{3}a}{2}\)

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

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

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

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

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