To find the radius of the circle in which an equilateral triangle is inscribed, we can follow a systematic approach. Let's break down the problem step-by-step. ### Step 1: Understanding the Geometry An equilateral triangle is a triangle where all three sides are equal, and all three angles are 60 degrees. When an equilateral triangle is inscribed in a circle, the circle is called the circumcircle, and the radius of this circle is known as the circumradius. ### Step 2: Formula for the Circumradius For an equilateral triangle, the circumradius \( R \) can be calculated using the formula: \[ R = \frac{a}{\sqrt{3}} \] where \( a \) is the length of a side of the triangle. ### Step 3: Plugging in the Values In this case, the side length \( a \) of the equilateral triangle is given as \( \sqrt{3} \) cm. We can substitute this value into the formula: \[ R = \frac{\sqrt{3}}{\sqrt{3}} = 1 \text{ cm} \] ### Step 4: Conclusion Thus, the radius of the circle in which the equilateral triangle is inscribed is **1 cm**. Therefore, the correct option is: **C. 1 cm** ### Step 5: Analyzing Other Options Now, let's examine the other options to understand why they are incorrect: - **A. 2/3 cm**: This value is too small. The circumradius for an equilateral triangle with a side length of \( \sqrt{3} \) cm cannot be less than 1 cm, as shown in our calculation. - **B. 2 cm**: This value is too large. The circumradius is directly proportional to the side length, and since the side length is \( \sqrt{3} \) cm, the circumradius cannot exceed this value. - **D. 3 cm**: This value is also too large for the same reasons as option B. The circumradius cannot exceed the side length of the triangle. ### Step 6: Common Pitfalls - **Misunderstanding the Formula**: It's crucial to remember that the circumradius formula for an equilateral triangle is specific and should not be confused with other types of triangles. - **Incorrect Substitution**: Ensure that the side length is correctly substituted into the formula. In this case, substituting \( a = \sqrt{3} \) correctly leads to the right answer. ### Revision Summary - The circumradius \( R \) of an equilateral triangle can be calculated using \( R = \frac{a}{\sqrt{3}} \). - For a triangle with side length \( \sqrt{3} \) cm, the circumradius is \( 1 \) cm. - The correct answer is **C. 1 cm**. - Always verify the dimensions and units when applying formulas to avoid common mistakes.