What is the circumference, in centimetres, of a circle whose area is 616 square centimetres?

Introduction / Context:
This problem connects two key formulas for a circle: the formula for area and the formula for circumference. You are given the area and asked to find the circumference, which requires determining the radius first. Such multi step geometry questions are very common in competitive exams.

Given Data / Assumptions:

• Area of the circle = 616 square centimetres.
• We need to find the circumference in centimetres.
• Use pi = 22 / 7 for calculations unless specified otherwise.

Concept / Approach:
The formulas involved are:
Area A = pi * r^2.
Circumference C = 2 * pi * r.
We first use the area to compute r^2 and then r. After finding the radius, we plug it into the circumference formula to obtain the final answer.

Step-by-Step Solution:
Step 1: Write the area formula: A = pi * r^2. Step 2: Substitute A = 616 and pi = 22 / 7. Step 3: So, 616 = (22 / 7) * r^2. Step 4: Multiply both sides by 7 to remove the denominator: 616 * 7 = 22 * r^2. Step 5: Compute 616 * 7 = 4312, so 4312 = 22 * r^2. Step 6: Divide both sides by 22: r^2 = 4312 / 22 = 196. Step 7: Take square root: r = sqrt(196) = 14 cm. Step 8: Now use circumference formula: C = 2 * pi * r = 2 * (22 / 7) * 14. Step 9: Simplify: 14 / 7 = 2, so C = 2 * 22 * 2 = 88 cm.

Verification / Alternative check:
We can check by reversing the process. If r = 14 cm, area A = pi * r^2 = (22 / 7) * 196 = 22 * 28 = 616 square centimetres, confirming the given area. The circumference with r = 14 cm is again 88 cm, matching our computed result.

Why Other Options Are Wrong:
44 cm: This is half of the correct circumference and would correspond to a radius of 7 cm, not 14 cm.
66 cm: This would correspond to a radius of approximately 10.5 cm and not give an area of 616 square centimetres.
132 cm: This is larger than required and corresponds to a radius greater than 14 cm.
22 cm: Much too small and clearly inconsistent with such a large area.

Common Pitfalls:
Students often mix up formulas for area and circumference or forget to take the square root when solving for the radius. Another issue is careless simplification of fractions when using pi = 22 / 7. Always proceed systematically: find r^2, find r, then compute the circumference carefully.

Final Answer:
The circumference of the circle is 88 cm.