how to find the area of an equilateral triangle

Area of Equilateral Triangle

The area of an equilateral triangle is the amount of space that an equilateral triangle covers in a 2-dimensional plane. An equilateral triangle is a triangle with all sides equal and all its angles measuring 60º. 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.

1.	What is the Area of an Equilateral Triangle?
2.	Area of Equilateral Triangle Formula
3.	Derivation of Area of Equilateral Triangle
4.	How to Find Area of Equilateral Triangle?
5.	FAQs on Area of Equilateral Triangle

What is the Area of an Equilateral Triangle?

The area of an equilateral triangle is defined as the region covered within the three sides of an equilateral side. It is expressed in square units. Some important units used to express the area of an equilateral triangle are in², m², cm², yd², etc. Let us understand the formula to calculate the area of an equilateral triangle and its derivation in the following sections.

Area of an Equilateral Triangle Formula

The equilateral triangle area formula is used to calculate the space occupied between the sides of the equilateral triangle in a plane. 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 general formula for the area of a triangle whose base and height are known is given as:

Area = 1/2 × base × height

While the formula to calculate the area of an equilateral triangle is given as,

Area = √3/4 × (side)² square units

area of equilateral triangle

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

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

Solution:

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.

Derivation of Area of an Equilateral Triangle Formula

In an equilateral triangle, all the sides are equal and all the internal angles are 60°. So, an equilateral triangle's area can be calculated if the length of one side is known. The formula to calculate the area of an equilateral triangle is given as,
Area of an equilateral triangle = (√3/4) × a² square units
where,
a = Length of each side of an equilateral triangle

The above formula to find the area of an equilateral triangle can be derived in the following ways:

Using the general area of a triangle formula
Using Heron's formula
Using trigonometry

Deriving Equilateral Triangle's Area Using Area of Triangle Formula

The formula used to calculate the area of an equilateral triangle can be derived using the general area of the triangle formula. To do so, we require the length of each side and the height of the equilateral triangle. We will thus calculate the height of an equilateral triangle in terms of the side length.

The formula for the area of an equilateral triangle comes out from the general formula of an equilateral triangle which is the area of the triangle is equal to ½ × base × height. Derivation for the formula of an equilateral triangle is given below.

Area of triangle = ½ × base × height

For finding the height of an equilateral triangle we use the Pythagoras theorem (hypotenuse²= base² + height²).

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)

So, the formula of height comes as ½ × (√3 × side) and further the area of the equilateral triangle becomes ¼(√3 × side²).

Deriving 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 is known. We do not use angles or other distances for finding the area of a triangle using Heron's formula.

area of an equilateral triangle by 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

Deriving Area of Equilateral Triangle With 2 Sides and Included Angle (SAS)

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)

area of an equilateral triangle with two sides and included angle

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

How to Find the Area of Equilateral Triangle?

The following steps can be followed to find the area of an equilateral triangle using the side length,

Step 1: Note the measure of the side length of the equilateral triangle.
Step 2: Apply the formula to calculate the equilateral triangle's area given as, A = (√3/4)a², where, a is the measure of the side length of the equilateral triangle.
Step 3: Express the answer with the desired unit.

Now, that we have learned the formula and method to calculate the area of the equilateral triangle, let us see few solved examples for better understanding.

Examples on Area of Equilateral Triangle

Example 1: An equilateral triangle signboard needs to be painted in red color. Each side of the signboard measures 8 in. Find the area to be painted red.

Solution:

The area of an equilateral triangle is given is,

Area of equilateral triangle = √3/4 × (Side)²

= √3/4 × 8²

= 16√3

Therefore, Area to be painted red = 16√3 inch²

Answer: Area to be painted red = 16√3 inch²
Example 2: Using the equilateral triangle area formula, calculate the area of an equilateral triangle whose each side is 12 in?

Solution:

To find: Area of an equilateral triangle

Given:

Side = 12 in

Using the equilateral triangle area formula,

Area = √3/4 × (Side)²

= √3/4 × (12)²

= 36√3 in²

Therefore, the area of an equilateral triangle area 36√3 in².

Answer: Area of an equilateral triangle area 36√3 in².

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Practice Questions on Area of Equilateral Triangle

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FAQs on Area of an Equilateral Triangle

What Is the Area of an Equilateral Triangle in Math?

The area of an equilateral triangle in math is just like the area of any other shape is the region encompassed or enclosed within the three sides of the equilateral triangle. It is expressed in square units or (unit)².

What Is the Formula of Equilateral Triangle Area?

We can calculate the area of an equilateral triangle given the length of each side. The formula of equilateral triangle area is equal to √3/4 times of square of the side length of the equilateral triangle.

☛ Also Check:

Triangle Formula
Equilateral Triangle Formulas
Area Formulas

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

The area of an equilateral triangle is √3/4 times (side)² of the equilateral triangle and the perimeter of an equilateral triangle is 3 times of a side of the equilateral triangle.

How Do You Calculate the Height Using Area of an Equilateral Triangle?

Given the area of an equilateral triangle, we can find the measure of each side using the formula, Area = √3/4 × (side)². For finding the height of an equilateral triangle from the side length, we use the Pythagoras theorem (hypotenuse²= base² + height²). So, the formula of height comes as ½ × (√3 × side).

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)a²where a is the length of the side of the equilateral triangle.

What is the Use of the Area of Equilateral Triangle Calculator?

Area of an equilateral triangle calculator is an online tool used to determine the area. This is the quickest mode to calculate the area of an equilateral triangle by providing an input value such as the length of the side. Try Cuemath's area of an equilateral triangle calculator now and calculate the area in a few seconds.

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

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-center of the triangle are the same points.