**Area of an Equilateral Triangle**

**Area of an Equilateral Triangle** is measured in unit^{2 }and the Equilateral triangles are also equi-angular, which denotes, all three internal angles are also equivalent to each other and the only value possible is 60° each.

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### Area of an Equilateral Triangle

** How To Find Area Of Equilateral Triangle?**

This is the Formula to Find Area of Equilateral Triangle as shown below:

Area = √3/4*a^{2}

Where ‘a’ is the length of any side of the triangle

Note that:√3/4 is constant that has the value of approximately 0.433, so the formula simplifies a little to

Approximate Area Of The Equilateral Triangle=0.433s^{s}

How To Find Area Of Right Angled Triangle: __Formula with Examples__

__For Example__:

**1:** Find the area of an equilateral triangle whose side is 7 cm?

**Solution:** Given, Side of the equilateral triangle = a = 7 cm

Area of an equilateral triangle =√3/4*a^{2}

= √3/4*7^{2}cm^{2 }= √3/4*49 cm^{2 }= 21.21762 cm^{2}

**2.** Find the area of an equilateral triangle whose side is 28 cm?

**Solution: **Given, Side of the equilateral triangle = a = 28 cm

The Area Of An Equilateral Triangle=√3/4*a^{2}

= √3/4*28^{2}cm^{2 }= √3/4*784 cm^{2 }= 339.48196cm^{2}

Formula to Calculate Area, Perimeter, Circumference: __Find Area of Circle__

** How to Calculate Derivation of Area?**

As we know that the Formula for Area of Equilateral Triangle is√3/4*a^{2}, then in terms of side length a can be derived directly using the following methods i.e.

- Pythagorean Theorem
- Trigonometry

** How to use Pythagorean Theorem to find Area of an Equilateral Triangle?**

The area of a triangle is half of one side ‘a’ times the height ‘h’ from that side:

A = 1/2*ah

The height of an equilateral triangle can be found using the Pythagorean Theorem is:

(a/2)^{2} + h^{2} = a^{2}

So that,

h= √3/4*a

And, replacing h into the area formula (1/2)ah gives the area formula for the equilateral triangle:

A = √3/4*a^{2}

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__How To Use Trigonometry To Find Area Of Equilateral Triangle__?

Now, in this condition, the area of a triangle with any two sides a and b, and an angle C between them is:

A = 1/2*ab*sinC

Each angle of an equilateral triangle is 60°, so

A = 1/2*ab*sin60^{0}

The sine of 600 is √3/2. Thus

A = 1/2ab *√3/2 = √3/4ab = √3/4a^{2}

__For Example:__

**1:** A farmer needs to replant a triangular section of crops that died unexpectedly. One side of the triangle measures 186 yards, another measures 205 yards, and the angle formed by these two sides is 148∘. What is the area of the section of crops that needs to be replanted?

**Solution:**

Use K=1/2bcsinA,

K=1/2(186)(205)sin148^{0}.

So, Area of a Equilateral Triangle that needs to be replaced is 10102.9 square yards.

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** How To Find Perimeter Of Equilateral Triangle When Area Is Given?**

Suppose, ΔABC is an equilateral triangle with area 55 then Perimeter is 33.8. Explanation is provided below:

__Explanation__:

To find out the perimeter of an equilateral triangle, firstly, we must find the length of the sides and this can be done by using the Area Of An Equilateral Triangle Formula:

Area = √3/4*a^{2}

Where ‘a’ is the side of the triangle, because the sides of the equilateral triangle are equal, the perimeter is equal to 3a.

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** Area Of Equilateral Triangle Proof**

Check here the Area Of Equilateral Triangle Formula Proof in following three steps as described below:

__Step 1__:

Since all the 3 sides of the triangle are same,

AB = BC = CA = a

__Step 2__:

Find the altitude of the △ABC.

First of all, Draw a perpendicular from point A to base BC, AD ⊥ BC

Now By using Pythagoras theoremIn △ADC

h^{2} = AC^{2} – DC^{2 }= a^{2} – (a/2)^{2} [Because, DC = a/2]

= a^{2} – a^{2}/4

h = √3a/2

__Step 3:__

We know that, Area of a triangle = 1/2 * Base * Height

=1/2*a*√3a/2

**= √3/4*a ^{2 }**

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