The problem involves an equilateral triangle with a perimeter of 642 centimeters. The three vertices of the triangle lie on a circle, and the radius of the circle is $w\sqrt{3}$ centimeters. We are tasked with finding the value of $w$.

Step-by-Step Solution:

1. Find the side length of the equilateral triangle: Since the perimeter of the equilateral triangle is 642 centimeters, and an equilateral triangle has three equal sides, we can calculate the length of each side: $\text{Side length} = \frac{642}{3} = 214 \text{ cm}.$

2. Relationship between the side of the equilateral triangle and the circumradius: For an equilateral triangle, the circumradius $R$ (the radius of the circumscribed circle) is related to the side length $s$ by the formula: $R = \frac{s}{\sqrt{3}}.$ Substituting the side length $s = 214$: $R = \frac{214}{\sqrt{3}} \text{ cm}.$

3. Set the given expression for the radius equal to this value: We are told that the radius of the circle is $w\sqrt{3}$ centimeters. Therefore: $w\sqrt{3} = \frac{214}{\sqrt{3}}.$ To solve for $w$, divide both sides by $\sqrt{3}$: $w = \frac{214}{3}.$ Simplifying this: $w = 71.33 \text{ cm}.$

Thus, the value of $w$ is approximately 71.33 centimeters.

Would you like more details or have any questions about this solution?

Here are some related questions for further practice:

1. What is the area of the equilateral triangle with side length 214 cm?
2. How do you derive the formula for the circumradius of an equilateral triangle?
3. If the radius of the circle was doubled, what would the new value of $w$ be?
4. Can this method be applied to other polygons inscribed in circles?
5. What is the difference between the circumradius and the inradius of an equilateral triangle?

Tip: Understanding the properties of regular polygons (like the circumradius formula) can make problems involving geometry much simpler to solve.