The area of a semicircle is 1925 cm². What is the perimeter (in cm) of this semicircle, including its diameter?

Introduction / Context:
This problem combines the area formula and perimeter formula for a semicircle. It tests your ability to work backwards from area to radius and then use the radius to compute the total perimeter, which includes both the curved arc and the straight diameter.

Given Data / Assumptions:

• The figure is a semicircle.
• Its area is 1925 cm².
• We are asked to find the perimeter (curved part + diameter) in cm.
• We will use π = 22 / 7 for calculations unless otherwise specified.
• Area of a full circle is πr²; area of a semicircle is (1 / 2) * πr².
• Perimeter of a semicircle is πr + 2r (half the circumference plus the diameter).

Concept / Approach:
First, we use the area of the semicircle to find the radius. The formula is: (1 / 2) * π * r² = 1925 We solve this equation for r. Once r is known, we compute the perimeter using: Perimeter = π * r + 2 * r This includes both the curved arc and the straight edge of the semicircle.

Step-by-Step Solution:
Step 1: Write the area equation for the semicircle: (1 / 2) * π * r² = 1925. Step 2: Substitute π = 22 / 7 to get (1 / 2) * (22 / 7) * r² = 1925. Step 3: Simplify (1 / 2) * (22 / 7) = 11 / 7, so (11 / 7) * r² = 1925. Step 4: Solve for r²: r² = 1925 * 7 / 11. Step 5: Compute 1925 / 11 = 175, so r² = 175 * 7 = 1225. Step 6: Therefore, r = √1225 = 35 cm. Step 7: Now compute the perimeter of the semicircle: Perimeter = πr + 2r. Step 8: Using π = 22 / 7, πr = (22 / 7) * 35 = 22 * 5 = 110 cm. Step 9: The diameter is 2r = 2 * 35 = 70 cm. Step 10: Total perimeter = 110 + 70 = 180 cm.

Verification / Alternative check:
We can double check by recomputing the area with r = 35 cm. Area of full circle is πr² = (22 / 7) * 35² = (22 / 7) * 1225 = 22 * 175 = 3850 cm². Half of this is 3850 / 2 = 1925 cm², which matches the given area. This confirms that r = 35 cm and thus perimeter 180 cm is consistent and correct.

Why Other Options Are Wrong:
80 cm and 160 cm are significantly smaller than the correct perimeter and would arise from incorrect radius calculations or from omitting either the arc or the diameter. 360 cm is about double the correct value and could come from mistakenly using the full circumference 2πr plus 2r. 140 cm corresponds to πr alone for r = 20, which does not satisfy the area condition of 1925 cm². None of these match the consistent radius and area relationship found above.

Common Pitfalls:
Students sometimes forget to include the diameter when calculating the perimeter of a semicircle and only compute πr. Another common mistake is misusing the area formula, for example by setting πr² = 1925 instead of (1 / 2) * πr². Careful attention to whether the figure is a full circle or a semicircle and using both arc and diameter in perimeter calculations will help avoid these errors.

Final Answer:
The perimeter of the semicircle is 180 cm.