Find the area of a triangle, whose sides are :
18 mm, 24 mm and 30 mm

Answer

To find the area of a triangle with sides measuring 18 mm, 24 mm, and 30 mm, we can use Heron's formula. Here’s a step-by-step solution:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( A = 18 \) mm
- \( B = 24 \) mm
- \( C = 30 \) mm

### Step 2: Calculate the semi-perimeter (S)
The semi-perimeter \( S \) of the triangle is calculated using the formula:
\[ S = \frac{A + B + C}{2} \]
Substituting the values:
\[ S = \frac{18 + 24 + 30}{2} = \frac{72}{2} = 36 \text{ mm} \]

### Step 3: Apply Heron's formula to find the area
Heron's formula for the area \( A \) of the triangle is given by:
\[ \text{Area} = \sqrt{S \times (S - A) \times (S - B) \times (S - C)} \]
Now, substituting the values:
\[ \text{Area} = \sqrt{36 \times (36 - 18) \times (36 - 24) \times (36 - 30)} \]
Calculating each term:
- \( S - A = 36 - 18 = 18 \)
- \( S - B = 36 - 24 = 12 \)
- \( S - C = 36 - 30 = 6 \)

So we have:
\[ \text{Area} = \sqrt{36 \times 18 \times 12 \times 6} \]

### Step 4: Simplify the expression
Now, we can simplify the expression:
\[ \text{Area} = \sqrt{36 \times 18 \times 12 \times 6} \]
Breaking down the numbers:
- \( 36 = 6 \times 6 \)
- \( 18 = 2 \times 3 \times 3 \)
- \( 12 = 2 \times 2 \times 3 \)
- \( 6 = 2 \times 3 \)

Combining these factors:
\[ 36 \times 18 \times 12 \times 6 = (6 \times 6) \times (2 \times 3 \times 3) \times (2 \times 2 \times 3) \times (2 \times 3) \]
Calculating the product:
\[ = 6^4 \times 2^4 \times 3^4 = 216 \text{ mm}^2 \]

### Final Answer
Thus, the area of the triangle is:
\[ \text{Area} = 216 \text{ mm}^2 \]

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