Find the area of a triangle whose sides are 12 cm, 16 cm and 20 cm.

Answer

To find the area of a triangle with sides measuring 12 cm, 16 cm, and 20 cm, we will use Heron's formula. Here’s the step-by-step solution:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( A = 12 \, \text{cm} \)
- \( B = 16 \, \text{cm} \)
- \( C = 20 \, \text{cm} \)

### Step 2: Calculate the semi-perimeter (S)
The semi-perimeter \( S \) is calculated using the formula:
\[ S = \frac{A + B + C}{2} \]
Substituting the values:
\[ S = \frac{12 + 16 + 20}{2} = \frac{48}{2} = 24 \, \text{cm} \]

### Step 3: Apply Heron's formula to find the area
Heron's formula for the area \( A \) of the triangle is given by:
\[ \text{Area} = \sqrt{S \cdot (S - A) \cdot (S - B) \cdot (S - C)} \]
Substituting the values of \( S \), \( A \), \( B \), and \( C \):
\[ \text{Area} = \sqrt{24 \cdot (24 - 12) \cdot (24 - 16) \cdot (24 - 20)} \]
Calculating the terms inside the square root:
\[ = \sqrt{24 \cdot 12 \cdot 8 \cdot 4} \]

### Step 4: Simplify the expression
Now, we will simplify \( 24 \cdot 12 \cdot 8 \cdot 4 \):
- First, calculate \( 24 \cdot 12 = 288 \)
- Next, calculate \( 8 \cdot 4 = 32 \)
- Now, multiply \( 288 \cdot 32 \):
\[ 288 \cdot 32 = 9216 \]

### Step 5: Calculate the square root
Now we find the square root of 9216:
\[ \text{Area} = \sqrt{9216} = 96 \, \text{cm}^2 \]

### Final Answer
Thus, the area of the triangle is \( 96 \, \text{cm}^2 \).

---