Find the area of an isosceles triangle whose equal sides are 5 cm each and base is 6 cm.

Answer

To find the area of an isosceles triangle with equal sides of 5 cm each and a base of 6 cm, 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 isosceles triangle be:
- \( a = 5 \) cm (one of the equal sides)
- \( b = 6 \) cm (the base)
- \( c = 5 \) cm (the other equal side)

### 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:
\[ s = \frac{5 + 6 + 5}{2} = \frac{16}{2} = 8 \text{ cm} \]

### Step 3: Apply Heron's formula
Heron's formula for the area \( A \) of the triangle is given by:
\[ A = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)} \]
Substituting the values of \( s \), \( a \), \( b \), and \( c \):
\[ A = \sqrt{8 \cdot (8 - 5) \cdot (8 - 6) \cdot (8 - 5)} \]
This simplifies to:
\[ A = \sqrt{8 \cdot 3 \cdot 2 \cdot 3} \]

### Step 4: Simplify the expression
Calculating the product inside the square root:
\[ A = \sqrt{8 \cdot 3 \cdot 2 \cdot 3} = \sqrt{8 \cdot 18} = \sqrt{144} \]
Since \( \sqrt{144} = 12 \), we find:
\[ A = 12 \text{ cm}^2 \]

### Conclusion
The area of the isosceles triangle is \( 12 \) cm².

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