The height of the equilateral triangle is 9 cm. What is the radius (in cm) of the circle circumscribing the three vertices?

Options

A. 3
B. 6
C. 9
D. 12

Answer

To find the radius of the circle circumscribing an equilateral triangle with a height of 9 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between the height and the side of the equilateral triangle**:
The height (h) of an equilateral triangle can be expressed in terms of its side length (A) using the formula:
\[ h = \frac{\sqrt{3}}{2} A \]
Given that the height \( h = 9 \) cm, we can set up the equation:
\[ 9 = \frac{\sqrt{3}}{2} A \]

2. **Solve for the side length (A)**:
Rearranging the equation to solve for \( A \):
\[ A = \frac{9 \times 2}{\sqrt{3}} = \frac{18}{\sqrt{3}} \text{ cm} \]

3. **Calculate the radius (R) of the circumscribing circle**:
The radius \( R \) of the circumscribing circle for an equilateral triangle can be calculated using the formula:
\[ R = \frac{A}{\sqrt{3}} \]
Substituting the value of \( A \):
\[ R = \frac{\frac{18}{\sqrt{3}}}{\sqrt{3}} = \frac{18}{3} = 6 \text{ cm} \]

4. **Conclusion**:
The radius of the circle circumscribing the three vertices of the equilateral triangle is \( 6 \) cm.

### Final Answer:
The radius of the circle is **6 cm**.