The radius of the incircle of triangle when sides are 18, 24 and 30 cms is

Options

A. 2 cm
B. 4 cm
C. 6 cm
D. 9 cm

Answer

To find the radius of the incircle of a triangle with sides 18 cm, 24 cm, and 30 cm, we can follow these steps:

### Step 1: Calculate the semi-perimeter (s) of the triangle
The semi-perimeter \( s \) is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
where \( a = 18 \, \text{cm} \), \( b = 24 \, \text{cm} \), and \( c = 30 \, \text{cm} \).

Substituting the values:
\[ s = \frac{18 + 24 + 30}{2} = \frac{72}{2} = 36 \, \text{cm} \]

### Step 2: Calculate the area (A) of the triangle
The area of the triangle can be calculated using Heron's formula:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values of \( s \), \( a \), \( b \), and \( c \):
\[ A = \sqrt{36(36 - 18)(36 - 24)(36 - 30)} \]
Calculating each term:
- \( s - a = 36 - 18 = 18 \)
- \( s - b = 36 - 24 = 12 \)
- \( s - c = 36 - 30 = 6 \)

Now substituting these values back into the area formula:
\[ A = \sqrt{36 \times 18 \times 12 \times 6} \]

### Step 3: Simplify the area calculation
Calculating the product:
\[ 36 \times 18 = 648 \]
\[ 648 \times 12 = 7776 \]
\[ 7776 \times 6 = 46656 \]
Now taking the square root:
\[ A = \sqrt{46656} = 216 \, \text{cm}^2 \]

### Step 4: Calculate the radius of the incircle (r)
The radius of the incircle \( r \) is given by the formula:
\[ r = \frac{A}{s} \]
Substituting the values of area \( A \) and semi-perimeter \( s \):
\[ r = \frac{216}{36} = 6 \, \text{cm} \]

### Final Answer
The radius of the incircle of the triangle is \( 6 \, \text{cm} \).

---