Find the area of of a triangle, whose sides are :
21 m, 28 m and 35 m

Answer

To find the area of a triangle with sides measuring 21 m, 28 m, and 35 m, we can use Heron's formula. Here’s a step-by-step solution:

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

**Calculation:**
\[ s = \frac{21 + 28 + 35}{2} = \frac{84}{2} = 42 \text{ m} \]

### Step 2: Apply Heron's formula to find the area (A)
Heron's formula states that the area \( A \) of a triangle can be calculated using the formula:
\[ A = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)} \]

**Substituting the values:**
- \( a = 21 \) m
- \( b = 28 \) m
- \( c = 35 \) m

**Calculating each term:**
\[ s - a = 42 - 21 = 21 \]
\[ s - b = 42 - 28 = 14 \]
\[ s - c = 42 - 35 = 7 \]

### Step 3: Substitute into Heron's formula
Now substitute \( s \), \( s - a \), \( s - b \), and \( s - c \) into the area formula:
\[ A = \sqrt{42 \cdot 21 \cdot 14 \cdot 7} \]

### Step 4: Calculate the product
First, calculate the product:
\[ 42 \cdot 21 = 882 \]
\[ 882 \cdot 14 = 12348 \]
\[ 12348 \cdot 7 = 86436 \]

### Step 5: Find the square root
Now, take the square root of the product:
\[ A = \sqrt{86436} = 294 \text{ m}^2 \]

### Final Answer
The area of the triangle is \( 294 \text{ m}^2 \).

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