If triangle ABC is an equilateral triangle of side 'a', its area will be ________.

To determine the area of an equilateral triangle, we can use the formula specifically designed for this type of triangle.

1. ### Break Down the Problem

We are given that triangle ABC is an equilateral triangle with each side of length $a$. Our goal is to find the area of this triangle.

2. ### Relevant Concepts

The formula for the area $A$ of an equilateral triangle with side length $a$ is given by: $A = \frac{\sqrt{3}}{4}a^2$

3. ### Analysis and Detail

Let’s analyze the formula:

• The area is a function of the square of the side length.
• The factor $\sqrt{3}$ comes from the properties of the triangle and relates to the height of the triangle derived using Pythagorean theorem.

4. ### Verify and Summarize

To verify, we can derive the area from first principles:

1. The height $h$ of the equilateral triangle can be calculated using the Pythagorean theorem: $h = \frac{\sqrt{3}}{2}a$.
2. The area can also be expressed as $A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times a \times h$.
3. Substituting $h$: $A = \frac{1}{2} \times a \times \frac{\sqrt{3}}{2}a = \frac{\sqrt{3}}{4}a^2$

Both methods agree, confirming the formula.

Final Answer

Thus, the area of triangle ABC will be: $\frac{\sqrt{3}}{4}a^2$