If the perimeter of a semicircle is 108 cm, find the radius of the semicircle in centimetres. (Use pi = 22/7.)

Introduction / Context:
This question involves the perimeter of a semicircle, which consists of a curved arc plus a straight diameter. Given the perimeter of the semicircle, we must work backwards to find the radius. This requires knowing and correctly applying the formula for the perimeter of a semicircle, using the given value of pi. It is a good test of algebraic manipulation with geometric formulas.

Given Data / Assumptions:

• Perimeter of semicircle P = 108 cm.
• Radius of semicircle = r cm (unknown).
• Perimeter of a semicircle = pi * r + 2 * r (arc length plus diameter).
• Use pi = 22/7.

Concept / Approach:
We set up the equation P = pi * r + 2r, plug in P = 108 and pi = 22/7, and then solve for r. To simplify, we combine terms in r by expressing them with a common denominator. After we have an equation of the form coefficient * r = 108, we can isolate r by division. The key is arranging and simplifying the fractional expression correctly.

Step-by-Step Solution:
Perimeter P = pi * r + 2r.Given P = 108 and pi = 22/7.So 108 = (22/7) * r + 2r.Write 2 as 14/7 so that both terms have denominator 7: 2r = (14/7) * r.Therefore 108 = (22/7) * r + (14/7) * r = (36/7) * r.So (36/7) * r = 108.Multiply both sides by 7: 36 * r = 108 * 7.Compute 108 * 7 = 756, so 36 * r = 756.Therefore r = 756 / 36 = 21 cm.

Verification / Alternative check:
Using r = 21 cm, we can recompute the perimeter to verify. Arc length of semicircle = pi * r = (22/7) * 21 = 22 * 3 = 66 cm. Diameter = 2r = 2 * 21 = 42 cm. Total perimeter P = 66 + 42 = 108 cm, which exactly matches the given perimeter. This confirms that r = 21 cm is consistent with the problem statement.

Why Other Options Are Wrong:
If r = 42 cm, the perimeter would be (22/7) * 42 + 84 = 132 + 84 = 216 cm, not 108 cm. For r = 28 cm, perimeter would be (22/7) * 28 + 56 = 88 + 56 = 144 cm. For r = 56 cm, perimeter would be far larger. A radius of 14 cm gives perimeter (22/7) * 14 + 28 = 44 + 28 = 72 cm. None of these equal 108 cm, so these options are incorrect.

Common Pitfalls:
A common error is to forget the straight diameter term and use only pi * r, which underestimates the perimeter. Others mistakenly use pi * 2r (full circumference) instead of pi * r, which overestimates it. Learners also sometimes mismanage the fraction 22/7, leading to incorrect algebra. Writing the formula clearly and handling fractions step by step avoids these mistakes.

Final Answer:
21