About Area of an Equilateral Triangle
What is an Equilateral Triangle?
An equilateral triangle is a triangle whose all sides and all angles are equal. Each angle of an equilateral triangle measures 60°. It is the smallest closed regular polygon. All equilateral triangles are similar.
Derivation of Equilateral Triangle Area Formula
Let’s first look at the area of an equilateral triangle: \(\frac{\sqrt3}{4}\)side\(^2\)
In the next step, we derive the equation for the area of an equilateral triangle.
Consider an equilateral triangle with side length ‘a’
Draw a perpendicular bisector of a side and name the perpendicular, also called the height of the triangle, length as ‘h’
Now we use the formula for the area of a triangle:
Area of triangle ABC = \(\frac{1}{2}\) x base x height
= \(\frac{1}{2}\) x a x h
Since triangle ADC is a right angle triangle use Pythagoras theorem to find the relation between a and h.
H\(^2\) = P\(^2\) + B\(^2\)
Here, AC\(^2\) = AD\(^2\) + DC\(^2\)
Substitute the corresponding values
\(\text{a}^2=\text{h}^2+(\frac{\text{a}}{2})^2\)
\(\text{a}^2=\text{h}^2+\frac{\text{a}^2}{4}\)
Find the value of h in terms of a
a\(^2~-~\frac{\text{a}^2}{4}\) = h\(^2\)
\(\frac{3}{4}\)a\(^2\) = h\(^2\)
\(\frac{\sqrt3}{2}\)a = h
Now substitute the value of h in area formula
Area of triangle ABC = \(\frac{1}{2}\) x a x \(\frac{\sqrt3}{2}\)a
Area of triangle ABC = \(\frac{\sqrt3}{4}\)a\(^2\)
So, the area of an equilateral triangle \(\frac{\sqrt3}{4}\)side\(^2.\)
Solved Examples
Example 1: The side length of the sign board is 2 feet. Assume the board is an equilateral triangular shape. What area of steel sheeting was used to create the sign board? Use 1.732 for \(\sqrt{3.}\)
Solution:
A = \(\frac{\sqrt3}{4}\)side\(^2\) [Formula for area of an equilateral triangle]
A = \(\frac{1.732}{4}\times2^2\) [Substitute 2 for side and 1.732 for \sqrt{3.}]
A = 1.732 feet\(^2\) [Simplify]
So, a 1.732 square feet of steel sheeting was used to create the sign board.
Example 2: Find the area of a triangle all of whose sides are 6 centimeters in length. Use 1.732 for \(\sqrt{3.}\)
Solution:
Since all sides of a given triangle are equal, it is an equilateral triangle. We can use the area of the equilateral triangle formula to find the area of the given triangle.
A = \(\frac{\sqrt3}{4}\)side\(^2\) [Formula for area of equilateral triangle]
A = \(\frac{1.732}{4}\times6^2\) [Substitute 6 for side and 1.732 for \sqrt{3.}]
A = 15.558 cm\(^2\) [Simplify]
So, the area of the triangle is 15.558 square centimeters.
Example 3: The area of an equilateral triangle is 21.217 square millimeters. Find the side length of a triangle?
Solution:
A = \(\frac{\sqrt3}{4}\)side\(^2\) [Formula for area of an equilateral triangle]
21.217 = \(\frac{1.732}{4}\times\) side\(^2\) [Substitute 21.217 for A and 1.732 for \sqrt{3.}]
\(21.217 \times4=1.732\times\) side\(^2\) [Multiply each side by 4]
\(\frac{84.868}{1.732}=\) side\(^2\) [Divide 1.732 by both sides]
49 = side\(^2\) [Divide]
\(\sqrt{49}=\sqrt{\text{side}^{2}}\) [Take positive square root on both sides]
7 = side
So, the side length of an equilateral triangle is 7 millimeters.
Each angle of an equilateral triangle is 60 degrees.
There are three sides in a triangle.
There are three interior angles in a triangle.
Angle sum property of triangle states that ‘the sum of the all interior angles of a triangle is equal to 180 degrees’.
Perimeter of an equilateral triangle is stated as ‘3a’, in which ‘a’ is the side length of an equilateral triangle.
Yes, all the sides of an equilateral triangle are equal.