Find the area of a triangular field whose sides are 78 m, 50 m and 112 m.

Answer

To find the area of a triangular field with sides measuring 78 m, 50 m, and 112 m, we can 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:
- \( a = 78 \, \text{m} \)
- \( b = 50 \, \text{m} \)
- \( c = 112 \, \text{m} \)

### 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:
\[ s = \frac{78 + 50 + 112}{2} = \frac{240}{2} = 120 \, \text{m} \]

### Step 3: Apply Heron's formula to find the area (A)
Heron's formula states that the area \( A \) of the triangle can be calculated as:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Now, we will calculate each term:
- \( s - a = 120 - 78 = 42 \)
- \( s - b = 120 - 50 = 70 \)
- \( s - c = 120 - 112 = 8 \)

### Step 4: Substitute the values into Heron's formula
Now substituting the values into the formula:
\[ A = \sqrt{120 \times 42 \times 70 \times 8} \]

### Step 5: Calculate the product
First, calculate the product inside the square root:
\[ 120 \times 42 = 5040 \]
\[ 5040 \times 70 = 352800 \]
\[ 352800 \times 8 = 2822400 \]

### Step 6: Find the square root
Now, we find the square root:
\[ A = \sqrt{2822400} \]
Calculating the square root gives:
\[ A \approx 1680 \, \text{m}^2 \]

### Final Answer
The area of the triangular field is approximately \( 1680 \, \text{m}^2 \).

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