The lengths of the three sides of a triangle are 13 cm, 14 cm and 15 cm, respectively. Find the area (in cm²) of the incircle of this triangle. [Use π = 3.14]

To find the area of the incircle of the triangle, we can follow a step-by-step approach using the properties of triangles and their incircles.

Step 1: Calculate the semi-perimeter of the triangle
The lengths of the three sides of the triangle are given as:
$a=13\text{cm}$
$b=14\text{cm}$
$c=15\text{cm}$
The semi-perimeter ($s$) is calculated as:
$s=\frac{a+b+c}{2}$
$s=\frac{13+14+15}{2}=\frac{42}{2}=21\text{cm}$

Step 2: Calculate the area of the triangle using Heron's Formula
Heron's formula for the area ($A$) of a triangle is given by:
$A=\sqrt{s(s-a)(s-b)(s-c)}$
Substituting the values of $s$, $a$, $b$, and $c$:
$A=\sqrt{21(21-13)(21-14)(21-15)}$
$A=\sqrt{21\times 8\times 7\times 6}$
Let us factorize the terms inside the square root to simplify easily:
$21=3\times 7$
$8=2^3$
$6=2\times 3$
So,
$A=\sqrt{(3\times 7)\times 2^3\times 7\times (2\times 3)}$
$A=\sqrt{2^4\times 3^2\times 7^2}$
$A=2^2\times 3\times 7=4\times 3\times 7=84{\text{cm}}^2$

Step 3: Find the inradius of the triangle
The relation between the area of the triangle ($A$), the inradius ($r$), and the semi-perimeter ($s$) is:
$r=\frac{A}{s}$
Substituting the calculated values of $A$ and $s$:
$r=\frac{84}{21}=4\text{cm}$

Step 4: Find the area of the incircle
The area of the incircle is given by the formula:
$\text{Area of incircle}=\pi r^2$
Using $\pi =3.14$ and $r=4\text{cm}$:
$\text{Area of incircle}=3.14\times 4^2$
$\text{Area of incircle}=3.14\times 16=50.24{\text{cm}}^2$

Therefore, the area of the incircle is 50.24 cm².