The perimeter of a semicircle (including its diameter) is 108 cm. Using π = 22/7, what is the radius of the semicircle in centimetres?

Introduction:
This question examines your ability to apply the formula for the perimeter of a semicircle and to rearrange it to find the radius. A semicircle consists of half the circumference of a full circle plus the diameter as the straight edge. You are given the total perimeter and must work backwards to find the radius.

Given Data / Assumptions:

• Perimeter of the semicircle (including the diameter) = 108 cm.
• Use π = 22/7.
• We must find the radius r in centimetres.
• The perimeter formula for a semicircle is known and includes both the curved part and the diameter.

Concept / Approach:
The perimeter P of a semicircle is given by: P = π * r + 2 * r, where π * r is half the circumference of the corresponding full circle and 2r is the diameter. We substitute P = 108 and π = 22/7, then solve the resulting linear equation for r.

Step-by-Step Solution:
Step 1: Write the formula for the perimeter of a semicircle: P = π * r + 2r. Step 2: Substitute P = 108 and π = 22/7: 108 = (22/7) * r + 2r. Step 3: Factor out r: 108 = r * (22/7 + 2). Step 4: Express 2 with denominator 7: 2 = 14/7, so (22/7 + 14/7) = 36/7. Step 5: Thus: 108 = r * (36/7). Step 6: Multiply both sides by 7: 108 * 7 = 36 * r, so 756 = 36r. Step 7: Solve for r: r = 756 / 36 = 21 cm.

Verification / Alternative check:
If r = 21 cm, then semicircular perimeter is: π * r + 2r = (22/7) * 21 + 2 * 21 = 22 * 3 + 42 = 66 + 42 = 108 cm, which matches the given perimeter. Therefore, the radius calculation is correct.

Why Other Options Are Wrong:

• 28 and 42: These values are larger than 21 and would give a perimeter greater than 108 cm.
• 14: This would produce a perimeter π * 14 + 28 which is (22/7) * 14 + 28 = 44 + 28 = 72 cm, not 108 cm.
• 56: This is far too large and would lead to a much bigger perimeter than 108 cm.

Common Pitfalls:
A typical mistake is to use the full circumference formula 2 * π * r instead of π * r for the semicircular arc, or to forget to include the diameter 2r in the total perimeter. Another common error is mishandling the fraction 22/7 while simplifying. Carefully keeping track of terms and denominators avoids these errors.

Final Answer:
The radius of the semicircle is 21 cm.