Understanding Equilateral Triangle Area

An equilateral triangle is a triangle where all three sides are equal in length, and all three angles are equal (each measuring 60 degrees). To find the area of an equilateral triangle, we use a specific formula that depends on the length of its side.

Let the side length of the equilateral triangle be denoted by 'a'.

Formula for Equilateral Triangle Area

The formula for the area of an equilateral triangle with side length 'a' is given by:

\(\text{Area} = \dfrac{\sqrt{3}}{4} a^2\) square units.

This formula is derived using the height of the triangle. The height of an equilateral triangle with side 'a' is \(\dfrac{\sqrt{3}}{2}a\). The area of any triangle is \(\dfrac{1}{2} \times \text{base} \times \text{height}\). For an equilateral triangle, the base is 'a' and the height is \(\dfrac{\sqrt{3}}{2}a\). Thus, the area is:

\(\text{Area} = \dfrac{1}{2} \times a \times \dfrac{\sqrt{3}}{2} a = \dfrac{\sqrt{3}}{4} a^2\)

Comparing Options for Area of Equilateral Triangle

Let's look at the given options for the area of an equilateral triangle:

• \(\dfrac{\sqrt{3}}{4}a^2\) sq. units
• \(2 a^2\) sq. units
• \(\sqrt{3} \ a^2\) sq. units
• \(\dfrac{\sqrt{3}}{2} a^2\) sq. units

Comparing these options with the standard formula for the area of an equilateral triangle, \(\dfrac{\sqrt{3}}{4} a^2\), we can see which one matches.

The first option, \(\dfrac{\sqrt{3}}{4}a^2\) sq. units, exactly matches the derived formula for the area of an equilateral triangle with side 'a'.

The other options do not represent the correct formula for the area of an equilateral triangle.

Therefore, the correct area of an equilateral triangle with side 'a' is \(\dfrac{\sqrt{3}}{4}a^2\) sq. units.