If area of a triangle is 81 `cm^(2)` and its semiperimeter is 27 cm then area of incircle of the triangle is

Options

A. `6pi" "cm^(2)`
B. `3pi" "cm^(2)`
C. `18pi" "cm^(2)`
D. `9pi" "cm^(2)`

Answer

To find the area of the incircle of a triangle given its area and semi-perimeter, we can follow these steps:

### Step 1: Identify the given values
- Area of the triangle (A) = 81 cm²
- Semi-perimeter (s) = 27 cm

### Step 2: Calculate the radius of the incircle (r)
The radius of the incircle can be calculated using the formula:
\[ r = \frac{A}{s} \]
Substituting the values we have:
\[ r = \frac{81 \, \text{cm}^2}{27 \, \text{cm}} \]

### Step 3: Simplify the calculation
Now, perform the division:
\[ r = \frac{81}{27} = 3 \, \text{cm} \]

### Step 4: Calculate the area of the incircle
The area of the incircle is given by the formula:
\[ \text{Area of incircle} = \pi r^2 \]
Substituting the value of r:
\[ \text{Area of incircle} = \pi (3 \, \text{cm})^2 \]
\[ \text{Area of incircle} = \pi \times 9 \, \text{cm}^2 \]
\[ \text{Area of incircle} = 9\pi \, \text{cm}^2 \]

### Final Answer
The area of the incircle of the triangle is \( 9\pi \, \text{cm}^2 \).

---