Explanation:

Introduction:
When an equilateral triangle is inscribed in a circle, it means that the three vertices of the triangle lie on the circumference of the circle. In such a scenario, we can find the ratio between the side of the equilateral triangle and the diameter of the circle.

Key Formula:
The key formula we need to use to solve this problem is the relationship between the side of an equilateral triangle and the diameter of the circle it is inscribed in.

Derivation:
Let's assume that the side of the equilateral triangle is 's' and the diameter of the circle is 'd'.

Step 1: Find the relationship between the side and the radius of the circle.
The radius of the circle is half of the diameter, so we can express it as:
radius (r) = d/2

Step 2: Find the relationship between the side and the radius of the circle.
In an equilateral triangle, all sides are equal. Therefore, the side (s) of the triangle is equal to the diameter of the circle inscribed in it.
s = d

Step 3: Find the relationship between the side and the radius of the circle.
Since the side (s) of the triangle is equal to the diameter of the circle, we can substitute s with d in the formula from Step 1:
radius (r) = s/2

Step 4: Find the relationship between the side and the radius of the circle.
We can now find the ratio between the side of the triangle and the radius of the circle:
s : r = s : (s/2) = 2 : 1

Step 5: Find the relationship between the side and the diameter of the circle.
Since the diameter is twice the radius, we can find the ratio between the side of the triangle and the diameter of the circle:
s : d = 2 : 1

Step 6: Simplify the ratio.
To simplify the ratio, we can multiply both sides by √3:
2√3 : √3 = √3 : 2

Conclusion:
The ratio of the side of the equilateral triangle to the diameter of the circle is √3 : 2, which corresponds to option B.