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? 

• A. 16
• B. None of the other answers
• C. \(12 + 2\pi\)
• D. \(16 + 2\pi\)
• E. \(20\pi\)

Hint:

For composite figures, carefully separate area and perimeter contributions of square and semicircle.

Answer 1

The Correct Option is C

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: \[ \boxed{12 + 2\pi} \]