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### Theory:

The perimeter of the sector:
The perimeter of the sector is defined as the sum of the two radii $\mathit{OA}\phantom{\rule{0.147em}{0ex}}\left(r\right)$, $\mathit{OB}\phantom{\rule{0.147em}{0ex}}\left(r\right)$ and the length of the arc $\stackrel{⌢}{\mathit{AB}\phantom{\rule{0.147em}{0ex}}}\left(l\right)$. It looks like a piece of a pie or a piece of pizza. Perimeter $$= r+r+l=2r+l$$ units.
The perimeter of a semicircle:
The perimeter of a semicircle is defined as the sum of the two radii $\mathit{OA}\phantom{\rule{0.147em}{0ex}}\left(r\right)$, $\mathit{OB}\phantom{\rule{0.147em}{0ex}}\left(r\right)$ and the length of the arc $\stackrel{⌢}{\mathit{AB}\phantom{\rule{0.147em}{0ex}}}\left(l\right)$. In this, we substitute the value of $$l =$$ $\mathrm{\pi }r$. Perimeter of a semicircle = $$l+2r$$ (where $$l=\pi r$$).

$$=\pi r + 2 r = (\pi + 2)r$$ units.
The perimeter of a circular quadrant:
The perimeter of a circular quadrant is defined as the sum of the two radii $\mathit{OA}\phantom{\rule{0.147em}{0ex}}\left(r\right)$, $\mathit{OB}\phantom{\rule{0.147em}{0ex}}\left(r\right)$ and the length of the arc $\stackrel{⌢}{\mathit{AB}\phantom{\rule{0.147em}{0ex}}}\left(l\right)$. In this, we substitute the value of $$l =$$ $\frac{\mathrm{\pi }r}{2}$. Perimeter of a quadrant = $$l+2r$$ (where $$l=\frac{\pi r}{2}$$).

$$=\frac{\pi r}{2} + 2r = \left(\frac{\pi}{2} + 2 \right) r$$ units.