Area of Equilateral Triangle

The area of an equilateral triangle is the amount of space enclosed within its three equal sides. For an equilateral triangle, where all three sides and all three internal angles are equal (each angle measuring 60 degrees), the area can be calculated using the formula where a is length of a side of Equilateral Triangle. An equilateral triangle is a triangle whose all side is equal to 60°.

👁 Area-of-Equilateral-Triangle

In simple words, Area of an equilateral triangle is the space occupied by an equilateral triangle. Let's know more about Area of Equilateral Triangle with formula, proof and examples.

Area of Equilateral Triangle

The area of an equilateral triangle is the amount of space enclosed within its three equal sides of an Equilateral Triangle. It is measured in square units. Area of an equilateral depends upon the length of a side of an equilateral triangle. Let's learn the formula used to define the area of an equilateral triangle.

👁 examples of equilateral triangles

Area of Equilateral Triangle Formula

Area of an equilateral triangle is the space occupied between the sides of the equilateral triangle in a plane. Below is the formula for finding the area of a triangle whose base and height are given is

Area =

👁 Area of Triangle

• If only sides of the triangle are given. Let an equilateral triangle of side 'a' be given then area of Equilateral Triangle is

👁 Area of Equilateral Triangle

Area of Equilateral Triangle Formula Proof

Let's calculate the area for a given equilateral triangle of side a. It is known that the area of a triangle is given as 1/2 × Base × Height.

👁 Equilateral Triangle's Area Derivation

Here the base is a. Let's find the height of this triangle in order to find the area. It can clearly be seen that the height can be found using the Pythagoras theorem since it is one of the sides of the right-angled triangle.

Applying Pythagoras' theorem,

h² + (a/2)² = a²

⇒ h² = (3a²/4)

⇒ h = √3a/2

Now the height of this equilateral triangle is known. Now, substitute this value of height into our formula, 

Area = 1/2 × Base × Height 

⇒ Area = 1/2 × a × √3a/2  =√3a²/4 

Area = √3a²/4

Derivation of Area of Equilateral Triangle using Trigonometry

Suppose the sides of a triangle are given, then the height can be calculated using the sine formula. Let the sides of a triangle ABC be a, b, and the angle corresponding to them be A, B, and C. Now, the height of a triangle is

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

Now, area of ABC = ½ × a × (b × sin C) 

⇒ area of ABC = ½ × b × (c × sin A)

⇒ area of ABC = ½ × c (a × sin B)

Since it is an equilateral triangle, A = B = C = 60° and a = b = c

⇒ Area = ½ × a × (a × Sin 60°) 

⇒ Area = ½ × a² × Sin 60°

⇒ Area = ½ × a² × √3/2 = √3a²/4 

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

Read More

• Trigonometry

Perimeter of the Equilateral Triangle

An equilateral triangle is a triangle with all three sides and the perimeter of any figure is the sum of all its sides. So, the perimeter of an equilateral triangle of side of length "a" is given by 

👁 Perimeter of the Equilateral Triangle

Must Read

• Perimeter of an equilateral triangle

Properties of Equilateral Triangle

An equilateral triangle is one triangle in which all three sides are equal. For an equilateral triangle PQR, PQ = QR = RP. A few important properties of an equilateral triangle are:

• All three sides are equal in an Equilateral Triangle.

• In an equilateral triangle, all three internal angles are equal to each other and their value is 60°.

• For an equilateral triangle, the median, angle bisector, and perpendicular all are the same.

• Ortho-centre and centroid of an equilateral triangle are the same points.

• In an equilateral triangle, there are three lines of symmetry and also 3rd order rotational symmetry as well.

• Area of an equilateral triangle is √3 a²/ 4.

• Perimeter of an equilateral triangle is 3a.

Must Read

• Area of Triangle
• Area of Square
• Area of Rhombus
• Area of Rectangle
• Area of Parallelogram
• Area of Circle

Solved Examples on Area of Equilateral triangle

Example 1: Find the area of the triangle whose all sides measure 4 units.

Solution:

As given all sides are of equal length hence, we can say that it is an equilateral triangle.

So we can apply the formula to directly find the area of this triangle.

Area = √3a²/4 = √3 × 4²/4 = 4√3 units²

Example 2: Find the perimeter of the triangle whose sides are given as 3 cm, 4 cm, and 5 cm.

Solution: 

Sum of all the sides of any triangle is the perimeter of triangle

Hence, the perimeter of this given triangle is (3 + 4 + 5) cm

i.e. Perimeter is 12 cm

Example 3: Find the height of the equilateral triangle whose side is 4 cm.

Solution:

The formula for the height is given by: h = √3a/2 

h = (√3 × 4)/2 = 2√3 cm

Hence the height of the triangle is 2√3 cm

Example 4: Find the perimeter and area of the equilateral triangle whose side is given as 4 cm.

Solution: 

Side (s) = 4 cm

For any equilateral triangle the perimeter is calculated as 3 × s

Primeter(P) = 3 × 4 = 12 cm

Area = √3a²/4 
        = √3(4)²/4
        = √3(16) / 4 cm²
Area = 4√3 cm²

Example 5: Find the area of an equilateral triangle when the perimeter is 18 cm.

Solution:

Perimeter of an equilateral triangle = 18 cm

Perimeter of the equilateral triangle = 3a

3a = 18, a = 6

The length of side is 6 cm.

Area, A = √3 a²/ 4 sq units

            = √3 (6)²/ 4 cm² ⇒ 36 √3 / 4

      = 9√3 cm²

Then area of the equilateral triangle is 9√3 cm²