What is the area of an equilateral triangle inscribed in a circle of unit radius?

### Concept & Inscribed Triangle Formula For an equilateral triangle inscribed in a circle of radius $R$, the side length $a$ is $R\sqrt{3}$. The area of an equilateral triangle is: $$\text{Area} = \frac{\sqrt{3}}{4} a^2$$ ### Step-by-Step Solution 1. **Find the side length ($a$):** The radius $R = 1$. $a = 1 \times \sqrt{3} = \sqrt{3}$ units. 2. **Calculate the Area:** $\text{Area} = \frac{\sqrt{3}}{4} (\sqrt{3})^2$ $\text{Area} = \frac{\sqrt{3}}{4} \times 3$ $\text{Area} = \frac{3\sqrt{3}}{4}$ sq. units. ### Exam Strategy & Shortcut You can also compute the area of the inscribed triangle directly using the circumradius formula: $\text{Area} = \frac{3\sqrt{3}}{4} R^2$. Since $R=1$, the area is simply the coefficient $\frac{3\sqrt{3}}{4}$. ### Common Pitfall Squaring $\sqrt{3}$ incorrectly, or forgetting the $1/4$ factor in the equilateral triangle area formula, which leads to choosing incorrect options like $3\sqrt{3}$. ### Final Answer Therefore, the correct answer is **$\frac{3\sqrt{3}}{4}$ sq. units**.