the area of an equilateral triangle is .

Options

A. `(sqrt(3))/(4)a^(2)sq` units
B. `2a^(2)` sq units
C. `3sqrt(3)a^(2)` sq units
D. `(sqrt(3))/(2)a^(2)` sq units

Answer

To find the area of an equilateral triangle, we can use the formula derived from Heron's formula. Here’s a step-by-step solution:

### Step 1: Understand the properties of an equilateral triangle
An equilateral triangle has all three sides equal. Let's denote the length of each side as \( a \).

### Step 2: Calculate the semi-perimeter
The semi-perimeter \( s \) of the triangle is calculated as:
\[ s = \frac{a + a + a}{2} = \frac{3a}{2} \]

### Step 3: Apply Heron’s formula
Heron’s formula for the area \( A \) of a triangle is given by:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Since all sides are equal in an equilateral triangle, we can substitute \( a \) for \( b \) and \( c \):
\[ A = \sqrt{s(s-a)(s-a)(s-a)} = \sqrt{s(s-a)^3} \]

### Step 4: Substitute the values into Heron's formula
Now, substituting \( s = \frac{3a}{2} \) and \( s-a = \frac{3a}{2} - a = \frac{a}{2} \):
\[ A = \sqrt{\frac{3a}{2} \left(\frac{a}{2}\right)^3} \]

### Step 5: Simplify the expression
Calculating \( \left(\frac{a}{2}\right)^3 \):
\[ \left(\frac{a}{2}\right)^3 = \frac{a^3}{8} \]
Now substituting back into the area formula:
\[ A = \sqrt{\frac{3a}{2} \cdot \frac{a^3}{8}} = \sqrt{\frac{3a^4}{16}} = \frac{a^2 \sqrt{3}}{4} \]

### Final Result
Thus, the area of an equilateral triangle with side length \( a \) is:
\[ A = \frac{\sqrt{3}}{4} a^2 \]