Question

In an equilateral triangle, each side has 6 cm then finds the area of the incircle.
a) \[3\pi c{{m}^{2}}\]
b) \[9\pi c{{m}^{2}}\]
c) \[12\pi c{{m}^{2}}\]
d) \[36\pi c{{m}^{2}}\]

Answer 1

Hint: To solve the question, we have to apply the formula of height of equilateral and draw perpendicular bisectors to the side of the equilateral triangle to analyse the area of the triangle in terms of inradius r. Thus, we can calculate the area of incircle using the value of r obtained.

Complete step by step answer:
Let r and h be the radius of incircle and height of the given equilateral triangle ABC.
We know that the formula for height of equilateral triangle ABC is given by \[h=\dfrac{\sqrt{3}a}{2}\]
Where A is the side of an equilateral triangle.
Given that each side of equilateral triangle ABC is equal to 6 cm.


Thus, the value of AB = BC = CA = a = 6 cm
By substituting the value in the formula of height of equilateral triangle, we get
\[h=\dfrac{\sqrt{3}}{2}\times 6=3\sqrt{3}cm\] ….. (1)

Let CD, BF, AE are perpendicular bisectors of sides of triangle ABC.
Let O be the centre of the incircle of the equilateral triangle ABC.
We can analyse from the diagram that the area of equilateral triangle ABC = Area of triangle AOB + Area of triangle AOC + Area of triangle COB
We know the formula for area of triangle = \[\dfrac{1}{2}\times \] base of triangle \[\times \] height of triangle
We can observe that,
The base of triangle AOB, triangle AOC, triangle COB is equal to the side of the equilateral triangle = a.
The height of triangle AOB, triangle AOC, triangle COB is equal to radius of incircle = r.
This implies that Area of triangle AOB = Area of triangle AOC = Area of triangle COB
Thus, the area of equilateral triangle ABC = 3(Area of triangle AOB)
Area of triangle AOB \[=\dfrac{1}{2}\times a\times r=\dfrac{ar}{2}\]
Thus, the area of equilateral triangle ABC \[=\dfrac{3ar}{2}\]
We know that area of equilateral triangle ABC of height h and base a \[=\dfrac{1}{2}\times a\times h\]

\[\Rightarrow \dfrac{3ar}{2}=\dfrac{ah}{2}\]
Thus, by comparing the comparing statements we get
h = 3r
By substituting the values from the equation (1) we get
\[3\sqrt{3}=3r\]
\[\Rightarrow r=\sqrt{3}\]
We know that area of circle with radius r is equal to \[\pi {{r}^{2}}\]
By substituting the value of r in the above formula we get
\[=\pi {{\left( \sqrt{3} \right)}^{2}}\]
\[=3\pi \]
Thus, the area of incircle is equal to \[=3\pi c{{m}^{2}}\]
Hence, option (a) is the right choice.

Note: The possibility of mistake is not able to analyse that the altitudes of the equilateral triangle can divide the triangle into equal triangles. The alternative method of solving the question is by using the direct formula for the radius of the incircle of the equilateral triangle with side a which is equal to \[r=\dfrac{2a}{\sqrt{3}}\]. Thus, by substituting the obtained value in the area of the circle we can obtain the required answer.