In a circle of radius 21 centimetres, an arc subtends a central angle of 30 degrees at the centre. Using the arc length formula, find the length of the arc in centimetres, taking pi = 22/7.

Introduction / Context:
This problem checks your understanding of the relationship between the central angle of a circle and the length of the corresponding arc. Arc length questions are standard in geometry and aptitude tests. The idea is to use the proportion of the angle to the full circle to find the fraction of the circumference that the arc represents.

Given Data / Assumptions:

• Radius of the circle r = 21 centimetres.
• Central angle theta = 30 degrees.
• We must find the length of the arc corresponding to this angle.
• Take pi = 22/7.
• The full circle has 360 degrees.

Concept / Approach:
The length of an arc is proportional to the angle it subtends at the centre. For a full circle, circumference = 2 * pi * r and angle = 360 degrees. For an arc with central angle theta, the arc length L is given by:
L = (theta / 360) * 2 * pi * r.
We substitute theta = 30 degrees, r = 21 centimetres, and pi = 22/7 into this formula and simplify step by step to obtain the arc length in centimetres.

Step-by-Step Solution:
Step 1: Use formula L = (theta / 360) * 2 * pi * r. With theta = 30 degrees, r = 21 cm, and pi = 22/7, we have L = (30 / 360) * 2 * (22/7) * 21. Step 2: Simplify the fraction 30 / 360 to 1 / 12. So L = (1 / 12) * 2 * (22/7) * 21. Step 3: Compute the full circumference term: 2 * (22/7) * 21. First, 21 divided by 7 is 3, so 2 * 22 * 3 = 132. Thus 2 * pi * r = 132 centimetres. Step 4: Now compute the fraction of the circumference: L = (1 / 12) * 132. 132 divided by 12 is 11. Therefore, the length of the arc is 11 centimetres.

Verification / Alternative check:
We can cross check by noting that 30 degrees is one twelfth of 360 degrees. So the arc length must be one twelfth of the full circumference. We already found the full circumference as 132 centimetres, so one twelfth of 132 is 11. The answer 11 cm is therefore perfectly consistent with both the formula and the proportional reasoning approach.

Why Other Options Are Wrong:
Option A (22 cm): This is exactly one sixth of the circumference and would correspond to a 60 degree central angle, not 30 degrees. Option B (16.5 cm): This value corresponds to an angle larger than 30 degrees and does not match the fraction 30/360. Option D (28 cm): This is too large and has no direct proportional link with the given angle. Option E (14 cm): This might tempt students who roughly guess, but it does not match the exact calculation.

Common Pitfalls:
Students sometimes forget to convert the angle to a fraction of 360 degrees and instead use 30 directly in the circumference formula. Another common mistake is to use pi * r instead of 2 * pi * r for the full circumference, or to mis-handle the fraction 30/360. Taking time to write the formula and simplify the angle ratio first makes the arithmetic straightforward and reduces errors.

Final Answer:
The length of the arc is 11 cm (11 centimetres).