What is the area (in square centimetres) of a circle whose circumference is 22 cm?

Introduction / Context:
This is another question linking circumference and area of a circle, but with a circumference that simplifies nicely when using π = 22/7. The goal is to find the radius from the given circumference and then compute the area with minimal calculation. Recognizing when 22/7 is convenient saves time and reduces arithmetic complexity in exams.

Given Data / Assumptions:

• Circumference C = 22 cm.
• Circumference formula: C = 2 * π * r.
• Area formula: A = π * r^2.
• Take π = 22/7, which is especially convenient here.
• We seek the area in square centimetres.

Concept / Approach:
We write 22 = 2 * π * r and substitute π = 22/7. This gives an equation for r that is easy to solve because the 22 terms cancel. Once r is known, we substitute into A = π * r^2 using the same π approximation. This yields a clean fractional value for the area, which can then be written as a decimal or left as a fraction depending on the options given.

Step-by-Step Solution:
Given C = 22 cm and C = 2 * π * r.Substitute π = 22/7: 22 = 2 * (22 / 7) * r.Simplify: 22 = (44 / 7) * r, so r = 22 * 7 / 44 = 154 / 44 = 3.5 cm.Now compute area A = π * r^2 = (22 / 7) * (3.5)^2.(3.5)^2 = 12.25, so A = (22 / 7) * 12.25 = 22 * 12.25 / 7 = 38.5 square centimetres.

Verification / Alternative check:
We can check the circumference from r = 3.5 cm to ensure consistency. Using C = 2 * π * r = 2 * (22 / 7) * 3.5 = (44 / 7) * 3.5. Since 3.5 = 7 / 2, the product is (44 / 7) * (7 / 2) = 44 / 2 = 22 cm, which matches the given circumference. This confirms that r = 3.5 cm and that the area 38.5 sq. cm is correct.

Why Other Options Are Wrong:
Option 77 sq. cm is double the correct area and would correspond to using radius 5 or misapplying the formula. Option 44 sq. cm is larger than the true area and does not fit with r = 3.5 cm. Option 88 sq. cm is far too large and more appropriate for a circle with much larger radius. Option 55 sq. cm is also too big and likely arises from rough approximation or ignoring the squared radius step.

Common Pitfalls:
Some students forget to substitute π as 22/7 and instead use 3.14, which complicates the arithmetic, though it can still give a close answer. Others solve incorrectly for r by not handling the fraction (44 / 7) properly. There is also confusion between circumference and area formulas, leading some to square the circumference directly or to omit the square on the radius in the area formula.

Final Answer:
The area of the circle is 38.5 square centimetres.