The perimeter of an equilateral triangle is 852 centimeters. The three vertices of the triangle lie on a circle. The radius of the circle is \( w \sqrt{3} \) centimeters. What is the value of \( w \) ?

To find the value of \( w \), we start with the perimeter of the equilateral triangle. Since the perimeter is 852 centimeters, each side of the triangle is: \[ \text{Side length} = \frac{852}{3} = 284 \text{ cm}. \] In an equilateral triangle, the relationship between the side length \( s \) and the radius \( R \) of the circumscribed circle (circumcircle) is given by: \[ R = \frac{s}{\sqrt{3}}. \] Substituting in our side length: \[ R = \frac{284}{\sqrt{3}}. \] We know that the radius is given as \( w \sqrt{3} \), so we can set them equal: \[ \frac{284}{\sqrt{3}} = w \sqrt{3}. \] To solve for \( w \), we multiply both sides by \( \sqrt{3} \): \[ 284 = w \cdot 3. \] Now divide by 3: \[ w = \frac{284}{3} \approx 94.67. \] Thus, the value of \( w \) is approximately \( 94.67 \).