The area of an equilateral triangle is :

Question

Accepted Answer

The area of an equilateral triangle can be calculated using the formula:

$$ \text{Area} = \frac{\sqrt{3}}{4} a^2 $$

where $ a $ is the length of a side of the equilateral triangle. This formula is derived from the general formula for the area of a triangle, $\frac{1}{2} \times \text{base} \times \text{height}$, and the properties of an equilateral triangle.

### Explanation for Option 1:

1. **Derivation of the Formula:**
   - In an equilateral triangle, all sides are equal, and all angles are 60 degrees.
   - To find the height, we can drop a perpendicular from one vertex to the opposite side, which bisects the side into two equal parts, each of length $\frac{a}{2}$.
   - Using the Pythagorean theorem in one of the resulting 30-60-90 right triangles, the height $ h $ can be calculated as:
     $$
     h = \sqrt{a^2 - \left(\frac{a}{2}\right)^2} = \sqrt{a^2 - \frac{a^2}{4}} = \sqrt{\frac{3a^2}{4}} = \frac{\sqrt{3}}{2}a
     $$
   - Substituting the height back into the area formula:
     $$
     \text{Area} = \frac{1}{2} \times a \times \frac{\sqrt{3}}{2}a = \frac{\sqrt{3}}{4}a^2
     $$
   - This confirms that the area of an equilateral triangle is indeed $\frac{\sqrt{3}}{4} a^2$ square units, which matches Option 1.

### Explanation for Why Other Options are Incorrect:

2. **Option 2: $2 a^2$ sq. units**
   - This option suggests that the area is simply twice the square of the side length, which does not take into account the geometric properties of an equilateral triangle. The formula for the area of an equilateral triangle involves the square root of 3, which is missing here.

3. **Option 3: $\sqrt{3} a^2$ sq. units**
   - This option incorrectly suggests that the area is directly proportional to $\sqrt{3}$ times the square of the side length. While $\sqrt{3}$ is part of the correct formula, it is not multiplied by $a^2$ directly but rather divided by 4, as shown in the correct formula.

4. **Option 4: $\frac{\sqrt{3}}{2} a^2$ sq. units**
   - This option is close but incorrect because it does not account for the division by 4, which is necessary to accurately calculate the area of an equilateral triangle. The correct formula divides by 4, not 2.

In conclusion, Option 1 is the correct answer because it accurately represents the formula for the area of an equilateral triangle, derived from its geometric properties.