An equilateral triangle is inscribed in a circle of radius `6` cm. Find its side.

Options

A. `6sqrt3`
B. `6/sqrt3`
C. `8sqrt3`
D. `2sqrt3`

Explanation

To find the side of an equilateral triangle inscribed in a circle of radius \(6\) cm, we can follow these steps:

### Step 1: Understand the Geometry
An equilateral triangle inscribed in a circle means that all its vertices touch the circle. The radius of the circle is the distance from the center of the circle to any vertex of the triangle.

### Step 2: Define the Triangle and Angles
Let the side of the equilateral triangle be denoted as \(a\). The angles of an equilateral triangle are \(60^\circ\) each. When we drop a perpendicular from the center of the circle to the midpoint of one side of the triangle, we create two \(30^\circ-60^\circ-90^\circ\) triangles.

### Step 3: Identify the Relevant Triangle
In one of the \(30^\circ-60^\circ-90^\circ\) triangles:
- The hypotenuse is the radius of the circle, which is \(6\) cm.
- The angle opposite to the side \(a/2\) is \(30^\circ\).
- The angle opposite to the height (perpendicular) is \(60^\circ\).

### Step 4: Use the Cosine Function
For the \(30^\circ\) angle, we can use the cosine function:
\[ \cos(30^\circ) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{a/2}{6} \]
From trigonometric values, we know:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]

### Step 5: Set Up the Equation
Substituting the cosine value into the equation gives:
\[ \frac{\sqrt{3}}{2} = \frac{a/2}{6} \]

### Step 6: Solve for \(a\)
Cross-multiplying to solve for \(a\):
\[ \sqrt{3} \cdot 6 = 2 \cdot \frac{a}{2} \]
\[ 6\sqrt{3} = a \]

### Step 7: Final Answer
Thus, the side length \(a\) of the equilateral triangle is:
\[ a = 6\sqrt{3} \text{ cm} \]