An equilateral triangle’s area is the amount of space it takes up in a two-dimensional plane. To refresh your memory, an equilateral triangle is one with all sides equal and all internal angles measuring 60 degrees. If the length of a side is known, the area of an equilateral triangle can be computed. Edureify brings to you complete information about the Equilateral Triangle.
Area of an Equilateral Triangle Formula
The formula for the area of an equilateral triangle is given:
Area of Equilateral Triangle (A) = (√3/4)a2
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”.
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 = ½ × base × height
Here, base = a, and height = h
Now, apply 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)
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 halves.
Now cut along the straight line and move the other half of the triangle to form the rectangle.
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, the area of ABC = ½ × a × (b . sin C) = ½ × b × (c . sin A) = ½ × c (a . sin B)
Now, since it is an equilateral triangle, A = B = C = 60°
And a = b = c
Area = ½ × a × (a . Sin 60°) = ½ × a2 × Sin 60° = ½ × a2 × √3/2
So, Area of Equilateral Triangle = (√3/4)a2
What is an Equilateral Triangle?
Scalene, equilateral, and isosceles triangles are the three primary types of triangles. All three sides of an equilateral triangle are equal, and all angles are 60 degrees. In an equilateral triangle, all of the angles are congruent.
Properties of Equilateral Triangle
All three sides of an equilateral triangle are the same length. It’s a specific case of the isosceles triangle with an equal third side. AB = BC = CA in an equilateral triangle ABC.
Some important properties of an equilateral triangle are:
- The triangle’s ortho-center and centroid are located at the same location.
- The median, angle bisector, and altitude for all sides of an equilateral triangle are all the same and constitute the equilateral triangle’s lines of symmetry.
- 3 a2/ 4 is the area of an equilateral triangle.
- An equilateral triangle’s perimeter is 3a.
- A triangle with all three sides equal is called an equilateral triangle.
- Equilateral triangles are also called equiangular. That means, all three internal angles are equal to each other and the only value possible is 60° 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 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 the perpendicular bisector due to congruence conditions.
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 a side is 4 cm.
Step 2: Find the area of an equilateral triangle using the formula.
Area, A = √3 a2/ 4 sq units
= √3 (4)2/ 4 cm2
= 4√3 cm2
Therefore, the area of the given equilateral triangle is 4√3 cm2
Question 2: What is the area of an equilateral triangle whose side is 8 cm?
Solution:
The area of the equilateral triangle = √3 a2/ 4
= √3 × (82)/ 4 cm2
= √3 × 16 cm2
= 16 √3 cm2
Frequently Asked Questions
Question No 1:- What is an Equilateral Triangle?
Ans:- 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.
Question No 2:- What does the Area of an Equilateral Triangle Mean?
Answer:- The area of an equilateral triangle is defined as the amount of space occupied by the equilateral triangle in the two-dimensional area.
Question No 3:- What is the Formula for the Area of an Equilateral Triangle?
Ans:- To calculate the area of an equilateral triangle, the following formula is used:
A = ¼(√3a2)
Question No 4:- What is the Formula for the Perimeter of an Equilateral Triangle?
Ans:- The formula to calculate the perimeter of an equilateral triangle is:
P = 3a
Question No 5:- How many sides are there in an equilateral triangle?
Ans:- There are three sides in an equilateral triangle.
