The area of an equilateral triangle is the amount of space that it occupies in a 2-dimensional plane. To recall, an equilateral triangle is a triangle in which all the sides are equal and the measure of all the internal angles is 60Â°. So, an equilateral triangle’s area can be calculated if the length of its side is known.

## Area of an Equilateral Triangle Formula

The formula for the area of an equilateral triangle is given as:

Area of Equilateral Triangle (A) = (âˆš3/4)a^{2} |

**Where a**** = length of sides**

Learn more about isosceles triangles, equilateral triangles and scalene triangles here.

## Derivation for Area of Equilateral Triangle

There are three methods to derive the formula for the area of equilateral triangles. They are:

- Using basic triangle formula
- Using rectangle construction
- Using trigonometry

### Deriving Area of Equilateral Triangle Using Basic Triangle Formula

Take an equilateral triangle of the side “a” units. Then draw a perpendicular bisector to the base of height “h”.

Now,

Area of Triangle = Â½ Ã— base Ã— height

Here, base = a, and height = h

Now, apply Pythagoras Theorem in the triangle.

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

â‡’ h^{2Â }= a^{2Â }â€“ (a^{2}/4)

â‡’ h^{2Â }= (3a^{2})/4

Or, h = Â½(âˆš3a)

Now, put the value of “h” in the area of the triangle equation.

Area of Triangle = Â½ Ã— base Ã— height

â‡’ A = Â½ Ã— a Ã— Â½(âˆš3a)

Or, **Area of Equilateral Triangle = Â¼(âˆš3a ^{2})**

### Deriving Area of Equilateral Triangle Using Rectangle Construction

Consider an equilateral triangle having sides equal to “a”.

- Now, draw a straight line from the top vertex of the triangle to the midpoint of the base of the triangle, thus, dividing the base into two equal halves.

- Now cut along the straight line and move the other half of the triangle to form the rectangle.

Here, the length of the equilateral triangle is considered to be ‘a’ and the height as ‘h’

So the area of an equilateral triangle = Area of a rectangle = Â½Ã—aÃ—h …………. (i)

Half of the rectangle is a right-angled triangle as it can be seen from the figure above.

Thereby, applying the Pythagoras Theorem:

â‡’ a^{2Â }= h^{2Â }+Â (a/2)^{2}

â‡’ h^{2Â }= (3/4)a^{2}

â‡’ h = (âˆš3/2)a â€¦â€¦â€¦â€¦â€¦(ii)

Substituting the value of (ii) in (i), we have:

**Area of an Equilateral Triangle**

=(Â½)Ã—aÃ—(âˆš3/2)a

=(âˆš3/4)a^{2}

### Deriving Area of Equilateral Triangle Using Trigonometry

If two sides of a triangle are given, then the height can be calculated using trigonometric functions. Now, the height of a triangle ABC will be-

h = b. Sin C = c. Sin A = a. Sin B

Now, area of ABC = Â½ Ã— a Ã— (b . sin C) = Â½ Ã— b Ã— (c . sin A) = Â½ Ã— c (a . sin B)

Now, since it is an equilateral triangle, A = B = C = 60Â°

And a = b = c

Area = Â½ Ã— a Ã— (a . Sin 60Â°) = Â½ Ã— a^{2} Ã— Sin 60Â° = Â½ Ã— a^{2} Ã— âˆš3/2

So, **Area of Equilateral Triangle = ****(âˆš3/4)a ^{2}**

Below is a brief recall about equilateral triangles:

### What is an Equilateral Triangle?

There are mainly three types of triangles which are scalene triangles, equilateral triangles and isosceles triangles. An equilateral triangle has all the three sides equal and all angles equal to 60Â°. All the angles in an equilateral triangle are congruent.

### Properties of Equilateral Triangle

An equilateral triangle is the one in which all three sides are equal. It is a special case of the isosceles triangle where the third side is also equal. In an equilateral triangle ABC, AB = BC = CA.

Some important properties of an equilateral triangle are:

- An equilateral triangle is a triangle in which all three sides are equal.
- Equilateral triangles also called equiangular. That means, all three internal angles are equal to each other and the only value possible is 60Â° each.
- It is a regular polygon with 3 sides.
- A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid.
- A triangle is equilateral if and only if any three of the smaller triangles have either the same perimeter or the same inradius.
- The area of an equilateral triangle is basically the amount of space occupied by an equilateral triangle.
- In an equilateral triangle, the median, angle bisector and perpendicular are all the same and can be simply termed as the perpendicular bisector due to congruence conditions.
- The ortho-centre and centroid of the triangle is the same point.
- In an equilateral triangle, median, angle bisector and altitude for all sides are all the same and are the lines of symmetry of the equilateral triangle.
- The area of an equilateral triangle is âˆš3 a
^{2}/ 4 - The perimeter of an equilateral triangle is 3a.

### Example Questions Using the Equilateral Triangle Area Formula

**Question 1**: **Find the area of an equilateral triangle whose perimeter is 12 cm.**

**Solution**:

Given: Perimeter of an equilateral triangle = 12 cm

As per formula: Perimeter of the equilateral triangle = 3a, where “a” is the side of the equilateral triangle.

**Step 1: **Find the side of an equilateral triangle using perimeter.

3a = 12

a = 4

Thus, the length of side is 4 cm.

**Step 2: **Find the area of an equilateral triangle using formula.

Area, A = âˆš3 a^{2}/ 4 sq units

= âˆš3 (4)^{2}/ 4Â cm^{2}

= 4âˆš3Â cm^{2}

Therefore, the area of the given equilateral triangle is 4âˆš3 cm^{2}

**Question 2**: **What is the area of an equilateral triangle whose side is 8Â cm?**

**Solution**:

The area of the equilateral triangle = âˆš3 a^{2}/ 4

= âˆš3 Ã— (8^{2})/ 4 cm^{2}

= âˆš3 Ã— 16 cm^{2}

= 16 âˆš3 cm^{2}

**Question 3:** **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 = âˆš3 a^{2}/ 4

= (âˆš3/4) Ã— 7^{2} cm^{2}

= (âˆš3/4) Ã— 49 cm^{2}

= 21.21762 cm^{2}

**Question 4: Find the area of an equilateral triangle whose side is 28 cm.**

**Solution:**

Given,

Side of the equilateral triangle (a) = 28 cm

We know,

Area of an equilateral triangle = âˆš3 a^{2}/ 4

= (âˆš3/4) Ã— 28^{2} cm^{2}

= (âˆš3/4) Ã— 784 cm^{2}

= 339.48196 cm^{2}

## Frequently Asked Questions

### What is an Equilateral Triangle?

An equilateral triangle can be defined as a special type of triangle whose all the sides and internal angles are equal. In an equilateral triangle, the measure of internal angles is 60 degrees.

### What does the Area of an Equilateral Triangle Mean?

The area of an equilateral triangle is defined as the amount of space occupied by the equilateral triangle in the two-dimensional area.

### What is the Formula for Area of Equilateral Triangle?

To calculate the area of an equilateral triangle, the following formula is used:

A = Â¼(âˆš3a^{2})

### What is the Formula for Perimeter of Equilateral Triangle?

The formula to calculate the perimeter of an equilateral triangle is:

P = 3a

WELL EXPLAINED

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