Find the area of a triangle whose sides are 26 cm, 28 cm and 30 cm.

Answer

To find the area of a triangle with sides measuring 26 cm, 28 cm, and 30 cm, 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 a 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.

Given:
- \( a = 26 \) cm
- \( b = 28 \) cm
- \( c = 30 \) cm

Now, substituting the values:
\[ s = \frac{26 + 28 + 30}{2} = \frac{84}{2} = 42 \text{ cm} \]

### Step 2: Apply Heron's formula to find the area (A)
Heron's formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)} \]

Substituting the values of \( s \), \( a \), \( b \), and \( c \):
\[ A = \sqrt{42 \cdot (42 - 26) \cdot (42 - 28) \cdot (42 - 30)} \]
Calculating the differences:
\[ A = \sqrt{42 \cdot (16) \cdot (14) \cdot (12)} \]

### Step 3: Simplify the expression
Now we will calculate the product inside the square root:
\[ A = \sqrt{42 \cdot 16 \cdot 14 \cdot 12} \]

Calculating each part:
- \( 42 = 2 \cdot 3 \cdot 7 \)
- \( 16 = 4^2 \)
- \( 14 = 2 \cdot 7 \)
- \( 12 = 3 \cdot 4 \)

Combining these:
\[ A = \sqrt{(2 \cdot 3 \cdot 7) \cdot (4^2) \cdot (2 \cdot 7) \cdot (3 \cdot 4)} \]
\[ = \sqrt{(2^3) \cdot (3^2) \cdot (7^2) \cdot (4^3)} \]

### Step 4: Calculate the square root
Now we can pair the factors to simplify:
\[ = \sqrt{(2^3) \cdot (3^2) \cdot (7^2) \cdot (4^3)} = \sqrt{(2^3) \cdot (3^2) \cdot (7^2) \cdot (2^6)} = \sqrt{(2^9) \cdot (3^2) \cdot (7^2)} \]
\[ = 2^{4.5} \cdot 3 \cdot 7 = 2^4 \cdot 2^{0.5} \cdot 3 \cdot 7 = 16 \cdot \sqrt{2} \cdot 21 \]

Calculating the area:
\[ A = 336 \text{ cm}^2 \]

### Final Answer
Thus, the area of the triangle is:
\[ \text{Area} = 336 \text{ cm}^2 \]

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