Area of Equilateral Triangle

The equilateral triangle area formula thus helps in calculating the space occupied between the sides of the equilateral triangle in a plane. The area of any shape is the number of unit squares that can fit into it. Here "unit" refers to one (1) and a unit square is a square with a side of 1 unit. Alternatively, the area of an equilateral triangle is the total amount of space it encloses in a 2-dimensional plane. An equilateral triangle is a triangle with all sides equal and all its angles measuring 60º.

In the given triangle ABC,

Area of ΔABC =  √3/4 × (side)² square units

Where,
AB = BC = CA = a (the length of equal sides of the triangle)

Thus, the formula for the area of the above equilateral triangle can be written as:

Area of equilateral triangle ΔABC = √3/4 × a² square units

How to find the area of an equilateral triangle with one side of 4 units?

Using the area of equilateral triangle formula: √3/4 × a² square units,

we will substitute the values of the side length. 

Therefore, the area of the equilateral triangle √3/4 × 4² = 4√3 square units. 

1.	Area of Equilateral Triangle Formula
2.	How to Find Area of Equilateral Triangle?
3.	Perimeter of Equilateral Triangle
4.	FAQs on Area of Equilateral Triangle

Area of an Equilateral Triangle Formula

The area equilateral triangle is the size of a two-dimensional surface. Alternatively, the area of a two-dimensional plane surface is a measure of the amount of space covered by it. Calculating areas of any plane is a very important skill used by many people in their work. For the equilateral triangle, we have the formula for its area. In a general triangle, finding the area of a triangle might be a little bit complicated for certain cases. But, finding the area of an equilateral triangle is quite an easy calculation.

The formula for the area of a triangle whose base and height are known is given as:

Area = 1/2 × Base × Height

While the area of an equilateral triangle is given as,

 Area  =  √3/4 × (side)² square units

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Area of Triangle = ½ × base × height

For finding the height of an equilateral triangle we use the Pythagoras theorem (hypotaneous² = base² + height²). So, the formula of height comes as  ½ × (√3 × side) and further the area of the equilateral triangle becomes ½ × side × (½ ×√3 × side).

Here, base = a, and height = h

Now, apply the Pythagoras theorem in the triangle.

a² = h² + (a/2)²

⇒ h² = a² – (a²/4)

⇒ h² = (3a²)/4

Or, h = ½(√3a)

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

Area of Triangle = ½ × base × height

⇒ A = ½ × a × ½(√3a)

Or, area of equilateral triangle = ¼(√3a²)

Area of Equilateral Triangle Using Heron's Formula

Heron's formula is used to find the area of a triangle when the length of the 3 sides of the triangle is known. In mathematics, Heron's formula named after hero of Alexandria. Who gives the area of any triangle when the length of all three sides are known. We do not use angles or other distances for finding the area of a triangle using Heron's formula.

Steps are given below for finding the area of a triangle:

Consider the triangle ABC with sides a, b and c.

Heron's formula to find the area of the triangle is:

Area = \(\sqrt {s(s - a)(s - b)(s - c)}\)

Where,

S is the semi-perimeter which is given by:

s = (a + b + c)/2

For equilateral triangle: a = b = c 

s = (a + a + a)/2

s = 3a/2

Now,
Area of equilateral triangle = \(\sqrt {s(s - a)(s - a)(s - a)}\)

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

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

= \(\sqrt{\frac{3a}{2}(\frac{{a^3}}{8}})\)

=\(\sqrt{({\frac{{3a^4}}{16}})}\)

Area of equilateral triangle = √3/4 × (side)² square units

For finding the area of a triangle side angle side (SAS) formula is a way to use the sine trigonometric function to calculate the height of a triangle and use that value to find the area of the triangle. There are three variations to the same formula based on which sides and included angle are given.

Consider a,b, and c are the different sides of a triangle.

• When sides 'b' and 'c' and included angle A is known, the area of the triangle is: 1/2 × bc × sin(A)
• When sides 'b' and 'a' and included angle B is known, the area of the triangle is: 1/2 × ab × sin(C)
• When sides 'a' and 'c' and included angle C is known, the area of the triangle is: 1/2 × ac × sin(B)

In an equilateral triangle, A = B = C = 60°

Sin A = Sin B = Sin C

Now, area of ABC = 1/2 × b × c × Sin(A) = 1/2 × a × b × Sin(C) = 1/2 × a × c × sin(B)

For equilateral triangle, a = b = c

Area = 1/2 × a × a × sin(C) = 1/2 × a²  × Sin(60°) = 1/2 × a² × √3/2

So, area of equilateral triangle = (√3/4)a²

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The word perimeter has been derived from the Greek word ‘peri’ meaning around, and ‘metron’ which means measure. Perimeter is the total length of a 2-D shape. The perimeter of a triangle means the sum of its boundary or all three sides. In the perimeter of a triangle, we add all three sides of a triangle like (a + b + c). In general, the perimeter is denoted as P. Let us see how to find the perimeter of an equilateral triangle.

The formula used to calculate the perimeter of a triangle is :

Perimeter = sum of the three sides

Let us see how to find the perimeter of an equilateral triangle.

Perimeter =  a + b + c.

a = b = c

a + a + a = 3a.

The perimeter of an equilateral triangle = 3a.

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1. What Is the Formula of Equilateral Triangle Area?

The formula of equilateral triangle area is equal to √3/4 times of side of the equilateral triangle.

2. How Do You Find the Area and Perimeter of an Equilateral Triangle?

The area of an equilateral triangle is √3/4 times of a side of the equilateral triangle and the perimeter of an equilateral triangle is 3 times of a side of the equilateral triangle.

3. How Do You Calculate the Height of an Equilateral Triangle?

For finding the height of an equilateral triangle we use the Pythagoras theorem (hypotaneous² = base² + height²). So, the formula of height comes as  ½ × (√3 × side).

4. What Is an Equilateral Triangle?

An equilateral triangle is a triangle with all sides equal and all its angles measuring 60º. The equilateral triangle area formula thus helps in calculating the space occupied between the sides

5. What Are the Properties of an Equilateral Triangle?

An equilateral triangle is a triangle with all sides equal and all its angles measuring 60º. Other properties that distinguish an equilateral triangle from other types of triangles are:

• For 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.
• For 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 centroid and ortho-centre of the triangle are the same point.

6. How Do Find the Sides of an Equilateral Triangle if the Area of an Equilateral Triangle Is Known?

If the area of an equilateral triangle is known, we put the given value in the following formula and solve for the length of the side:

Area of Equilateral Triangle = (√3/4)a²

Where a is the length of the side of the equilateral triangle.

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