Question

Find the area of an equilateral triangle of side “a” using heron’s formula.

Answer 1

Hint: Heron’s formula was created by Heron of Alexandria. It is also known as Heron's formula. This formula is most widely used where all the sides of the triangle are known and without knowing its angles. First here we will find the perimeter of the triangle and then its area by using Heron’s formula - $ A = \sqrt {s(s - a)(s - b)(s - c)} $

Complete step-by-step answer:

As shown in the above figure, an equilateral triangle has all its side equal.
Given that the length of the equilateral triangle $ = a $
 $ \Rightarrow AB = BC = CA = a $
Now, the semi-perimeter of the triangle is equal to the sum of all the three sides divided by two.
 $ s = \dfrac{1}{2}(AB + BC + CA) $
Place the value in the above equation –
  $
  s = \dfrac{1}{2}(a + a + a) \\
  s = \dfrac{{3a}}{2}\;{\text{ }}....{\text{ (1)}} \\
  $
Now, using the Heron’s formula –
 $ A = \sqrt {s(s - a)(s - b)(s - c)} $
Where “a” is the measure of the side “BC”
             “b” is the measure of the side “AC”
             “c” is the measure of the side “AB”
Now, place the values in the above equation –
 $ \Rightarrow A = \sqrt {\dfrac{{3a}}{2}(\dfrac{{3a}}{2} - a)(\dfrac{{3a}}{2} - a)(\dfrac{{3a}}{2} - a)} $
 Simplify the above equation –
 $ \Rightarrow A = \sqrt {\dfrac{{3a}}{2}(\dfrac{a}{2})(\dfrac{a}{2})(\dfrac{a}{2})} $
Rearranging the above terms-
 $ \Rightarrow A = \sqrt {\dfrac{{3\underline { \times a \times a} \times \underline {a \times a} }}{{\underline {2 \times 2} \times \underline {2 \times 2} }}} $
When the same number is multiplied twice, it is written as the square of that number.
 $ \Rightarrow A = \sqrt {\dfrac{{3 \times {a^2} \times {a^2}}}{{{2^2} \times {2^2}}}} $
By property – square and square-root cancel each other.
 $ \Rightarrow A = \dfrac{{\sqrt 3 a \times a}}{{2 \times 2}} $
Simplify above equation –
 $ \Rightarrow A = \dfrac{{\sqrt 3 {a^2}}}{4} $ Square units

Note: Heron’s formula is widely used where all the sides of the triangle are known and where the measure of its angle is not required. Simplify the equations placing the values and apply the square and square-root concepts carefully.
Heron’s formula is widely used in real and practical life, where the shape of the land is not always in the regular shape.