Why does $\pi$ cancel out when converting angle measures from radians to degrees?

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Can someone tell me why $\pi$ cancels out from this fraction when converting radians to degrees?

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$1$ rad $= \frac {180}{\pi}$ degrees, so $\frac {7 \pi}{8}$ radians = $\frac {7 \pi}{8} \cdot \frac {180}{\pi}$ degrees. The $\pi$ gets canceled with part of the conversion factor.