If side of an equilateral triangle is 'a', then area of this triangle is equal to :

Question

Accepted Answer

To determine the area of an equilateral triangle with side length 'a', we use the formula for the area of an equilateral triangle:

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

This formula is derived from the general formula for the area of a triangle, which is $\frac{1}{2} \times \text{base} \times \text{height}$. For an equilateral triangle, the height can be found using the Pythagorean theorem. If we drop a perpendicular from one vertex to the opposite side (which acts as the base), it divides the equilateral triangle into two 30-60-90 right triangles.

In a 30-60-90 triangle, the sides are in the ratio $1 : \sqrt{3} : 2$. Therefore, the height (opposite the 60-degree angle) is $\frac{\sqrt{3}}{2}a$, where 'a' is the hypotenuse (the side of the equilateral triangle).

Substituting the height into the area formula:

$$ \text{Area} = \frac{1}{2} \times a \times \frac{\sqrt{3}}{2}a = \frac{\sqrt{3}}{4} a^2 $$

This matches Option 3: $\frac{\sqrt{3}a^2}{4}$.

Now, let's explain why the other options are incorrect:

1. **Option 1: $\frac{3a^2}{2}$** - This option does not include the $\sqrt{3}$ factor, which is essential for the area calculation of an equilateral triangle. The formula $\frac{3a^2}{2}$ does not correspond to any standard geometric formula for the area of a triangle.

2. **Option 2: $\frac{\sqrt{3}a^2}{2}$** - This option is incorrect because it overestimates the area. The correct formula divides by 4, not 2, as the height calculation involves a division by 2 when using the Pythagorean theorem.

3. **Option 4: $\sqrt{3}a^2$** - This option is incorrect because it does not account for the division by 4, which is necessary to correctly calculate the area of an equilateral triangle. The formula $\sqrt{3}a^2$ would result in an area that is four times too large.

In conclusion, Option 3 is the correct answer because it accurately reflects the derived formula for the area of an equilateral triangle, taking into account the necessary geometric properties and calculations.