To find the radius of the circle, we need to understand the relationship between the circumference, diameter, and radius of a circle.

The circumference of a circle is the distance around its outer edge. It is calculated using the formula C = 2πr, where C represents the circumference and r represents the radius.

The diameter of a circle is the distance across its center, passing through any two points on the circle. It is calculated using the formula D = 2r, where D represents the diameter and r represents the radius.

Let's use these formulas to solve the problem step by step.

Step 1: Let's assume the radius of the circle is 'r' cm.

Step 2: According to the given information, the circumference of the circle exceeds its diameter by 18 cm. Mathematically, we can represent this as:
C = D + 18

Step 3: Substitute the formulas for circumference and diameter into the equation:
2πr = 2r + 18

Step 4: Simplify the equation by combining like terms:
2πr - 2r = 18

Step 5: Factor out the common term '2r' on the left side:
2r(π - 1) = 18

Step 6: Divide both sides of the equation by (π - 1) to solve for 'r':
2r = 18 / (π - 1)

Step 7: Divide both sides of the equation by 2 to isolate 'r':
r = (18 / (π - 1)) / 2

Step 8: Use a calculator to evaluate the right side of the equation:
r ≈ 4.2 cm

Therefore, the radius of the circle is approximately 4.2 cm. So, the correct answer is option C.