There are three types of triangles based on the length of the sides. They are as follows:
Equilateral triangle
Isosceles triangle, and
Scalene triangle.
Equilateral Triangle:
An equilateral triangle is a triangle whose all three sides are equal and each interior angle measures 60o.
Isosceles Triangle:
An isosceles triangle is a triangle whose two sides are equal.
Scalene Triangle:
A scalene triangle is a triangle whose all three sides are unequal.
Area of Equilateral Triangle
The area of an equilateral triangle is the amount of space that it occupies in a 2-dimensional surface.
So, the area of an equilateral triangle can be calculated if the length of its side is known.
The formula for the area of an equilateral triangle each of whose sides are ‘a’ unit is given by:
Area of an equilateral triangle =\[\left( {\frac{{\sqrt 3 }}{4}} \right) \times {a^2}\].
We can also use the decimal value of \[\sqrt 3 \] to simplify our calculations. Where \[\sqrt 3 \]= 1.732
Derivation of the Area of An Equilateral Triangle
The formula for the area of an equilateral triangle can be derived using any of the following two methods.
Using the basic area of a triangle formula
Using Heron’s formula
METHOD: 1
Deriving area of an equilateral triangle using the basic area of a triangle formula
Consider the following equilateral triangle ABC, whose each side is of length ‘a’ unit.
Since we know that the altitude from vertex A to opposite side BC of an equilateral triangle bisects it at point D.
So, BD = DC = \[\frac{a}{2}\] unit.
We can determine the length of altitude AD by using Pythagoras theorem.
If triangle ADC is Right-angled at angle D. Then, hypotenuse = AC, altitude = AD and base = DC.
According to Pythagoras theorem,
AC² = AD² + DC²
⇒ AD² = AC² - DC²
⇒ AD² = a² - (\[\frac{a}{2}\])² = a² - \[\frac{a^{2}}{4}\] = \[\frac{3}{4}\]a²
⇒AD = h = \[\frac{\sqrt{3}}{2}\] a
Now, area of equilateral ΔABC = \[\frac{1}{2}\] x Base x Height
= \[\frac{1}{2}\] x BC x AD
= \[\frac{1}{2}\] x a x h
= \[\frac{1}{2}\] x a x \[\frac{\sqrt{3}}{2}\] a
= \[\frac{\sqrt{3}}{4}\] a²
METHOD: 2
Deriving area of an equilateral triangle using Heron’s Formula
Heron’s Formula to find area of triangle = \[\sqrt {s\left( {s - a} \right)\left( {s - b} \right)\left( {s - c} \right)} \]
Where, s = semi-perimeter of \[\Delta ABC = \frac{{perimeter{\text{ }}of{\text{ }}\Delta ABC}}{2} = \frac{{a + b + c}}{2}\]
And ‘a’, ‘b’ and ‘c’ are the sides of ∆ABC
Perimeter of an equilateral triangle = sum of all sides of triangle = a + a + a = 3a
s = semi-perimeter of equilateral ∆ABC = 3a2
Since, In equilateral triangle each side has length measure ‘a’. i.e. length measures ‘a’,’b’ and ‘c’ are all ‘a’ units.
Putting the values of side length measures in Heron’s Formula, we get:
Area of equilateral \[\Delta ABC = \sqrt {\frac{{3a}}{2}\left( {\frac{{3a}}{2} - a} \right)\left( {\frac{{3a}}{2} - a} \right)\left( {\frac{{3a}}{2} - a} \right)} \]
\[ = \sqrt {\frac{{3a}}{2} \times \frac{a}{2} \times \frac{a}{2} \times \frac{a}{2}} \]
\[ = \frac{{\sqrt 3 }}{4}{a^2}\]
Solved Examples:
Q.1. Find the Area of an Equilateral Triangle Each of Whose Sides are 6 cm.
Ans.
Given each side of an equilateral triangle = a = 6 cm.
The formula of area of equilateral triangle \[ = \frac{{\sqrt 3 }}{4}{a^2}\]
\[ = \frac{{\sqrt 3 }}{4} \times {\left( 6 \right)^2}c{m^2}\]
\[ = 9\sqrt 3 c{m^2}\]
Q.2. The Perimeter of an Equilateral Triangle is 9 cm. Find its Altitude.
Ans.
The perimeter of an equilateral triangle is given by ‘3a’.
⇒ 3a = 9 cm
⇒ a = \[\frac{9}{3}\] = 3 cm
So, each side of the equilateral triangle measures 3 cm.
The altitude of an equilateral triangle is given by
\[h = \frac{{\sqrt 3 }}{2} \times \left( 3 \right)\]
Altitude \[h = \frac{{3\sqrt 3 }}{2}cm\]
Q.3. Find the Perimeter of an Equilateral Triangle with Area \[9\sqrt 3 {\text{ }}c{m^2}\].
Ans.
Area of equilateral triangle = \[\frac{\sqrt{3}}{4}\] a²
⇒ \[\frac{\sqrt{3}}{4}\] a² = 9 \[\sqrt{3}\]
⇒ a² = 36
⇒ a = 6 cm
Now, the perimeter of an equilateral triangle = 3a
= 3 x 6 cm
= 18 cm
Q.4. A Field is in the Form of an Equilateral Triangle with Side Length 12 m. Find the Cost of Fencing its Boundary at the Rate of Rs 50 per Meter.
Ans.
The length of the boundary of the field is found by obtaining the perimeter of an equilateral triangle.
The perimeter of an equilateral triangle = 3a
= 3 \[ \times \]12 m
= 36 m
⇒ The length of the boundary of Equilateral field = 36 m
So, the cost of fencing its boundary at the rate of Rs 50 per meter = Rs (36 \[ \times \] 50)
= Rs 1800
