Question:
The diagram represents a square \(ABCD\) with a semi-circle directly attached to its side. If the area of the figure is \(16 + 2\pi\), what is its outer perimeter?
The diagram represents a square \(ABCD\) with a semi-circle directly attached to its side. If the area of the figure is \(16 + 2\pi\), what is its outer perimeter?
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For composite figures, carefully separate area and perimeter contributions of square and semicircle.
Updated On: Oct 3, 2025
- 16
- None of the other answers
- \(12 + 2\pi\)
- \(16 + 2\pi\)
- \(20\pi\)
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The Correct Option is C
Solution and Explanation
Step 1: Let side of square = \(a\).
Area of square = \(a^2\). Radius of semicircle = \(a/2\). Step 2: Total area.
\[ a^2 + \tfrac{1}{2}\pi \left(\frac{a}{2}\right)^2 = 16 + 2\pi \] \[ a^2 + \frac{\pi a^2}{8} = 16 + 2\pi \] Step 3: Solve for \(a\).
This simplifies to \(a=4\). Step 4: Perimeter.
Perimeter = 3 sides of square + semicircle arc. \[ = 3a + \pi \cdot \frac{a}{2} = 12 + 2\pi \] Final Answer: \[ \boxed{12 + 2\pi} \]
Area of square = \(a^2\). Radius of semicircle = \(a/2\). Step 2: Total area.
\[ a^2 + \tfrac{1}{2}\pi \left(\frac{a}{2}\right)^2 = 16 + 2\pi \] \[ a^2 + \frac{\pi a^2}{8} = 16 + 2\pi \] Step 3: Solve for \(a\).
This simplifies to \(a=4\). Step 4: Perimeter.
Perimeter = 3 sides of square + semicircle arc. \[ = 3a + \pi \cdot \frac{a}{2} = 12 + 2\pi \] Final Answer: \[ \boxed{12 + 2\pi} \]
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