Find the area of an equilateral triangle whose sides are 4cm each.

Answer

To find the area of an equilateral triangle with sides of 4 cm each using Heron's formula, follow these steps:

### Step 1: Identify the side length
Given that the side length \( a = b = c = 4 \) cm.

### Step 2: Calculate the semi-perimeter
The semi-perimeter \( s \) of a triangle is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{4 + 4 + 4}{2} = \frac{12}{2} = 6 \text{ cm} \]

### Step 3: Apply Heron's formula
Heron's formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values of \( s \), \( a \), \( b \), and \( c \):
\[ A = \sqrt{6 \times (6 - 4) \times (6 - 4) \times (6 - 4)} \]
This simplifies to:
\[ A = \sqrt{6 \times 2 \times 2 \times 2} \]

### Step 4: Simplify the expression
Calculating the product inside the square root:
\[ A = \sqrt{6 \times 2^3} = \sqrt{6 \times 8} = \sqrt{48} \]
Now, we can simplify \( \sqrt{48} \):
\[ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} \]

### Step 5: Final result
Thus, the area of the equilateral triangle is:
\[ A = 4\sqrt{3} \text{ cm}^2 \]

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