Question

What is $5\pi $ in degrees?

Answer 1

Hint: We know that the relationship between radian and degrees is that, $1\text{ radian}=\dfrac{180}{\pi }\text{ degrees}$.
Thus, to convert radians into degrees, we have $\left( \dfrac{{{180}^{\circ }}}{\pi } \right)$ as the conversion factor. Thus, to convert $5\pi $ in degrees, we need to simply multiply $5\pi $ with the conversion factor $\left( \dfrac{{{180}^{\circ }}}{\pi } \right)$ .

Complete step by step solution:
We know that measurement of angles is mainly done in the following two units. These units are degrees and radians. These units are related to each other by the following relationships:
1 radian is same as $\dfrac{180}{\pi }$ degrees, that is,
$1\text{ radian}=\dfrac{180}{\pi }\text{ degrees}$.
And, 1 degree is same as $\dfrac{\pi }{180}$ radians, that is,
$1\text{ degree}=\dfrac{\pi }{180}\text{ radians}$.
In our question, we are given an angle whose measurement is $5\pi $ radians and we need to convert this angle into degrees.
We are now very well aware that the relationship between a radian and a degree is
$1\text{ radian}=\dfrac{180}{\pi }\text{ degrees}$
Hence, we have $\left( \dfrac{180}{\pi } \right)$ as the conversion factor.
So, to convert $5\pi $ radians into degrees, we just need to multiply $5\pi $ with our conversion factor $\left( \dfrac{180}{\pi } \right)$ . Thus, we get the following equation,
$\text{5}\pi \text{ radians}=\left( \dfrac{180}{\pi }\times 5\pi \right)\text{ degrees}$
We can now cancel the $\pi $ term from the denominator with the $\pi $ term in $5\pi $ . Thus, we get
$\text{5}\pi \text{ radians}=\left( 180\times 5 \right)\text{ degrees}$
Therefore, we can also express this as,
$\text{5}\pi \text{ radians}=900\text{ degrees}$
Hence, we can see that $5\pi $ radians are equal to 900 degrees.

Note: We can also solve the above problem using unitary method. We just need to remember that
$\pi \text{ radians}={{180}^{\circ }}$
So now, by unitary method, we can easily say that $\text{5}\pi \text{ radians}=\left( 5\times {{180}^{\circ }} \right)$
If we rewrite the above equation, we get $\text{5}\pi \text{ radians}=900$ degrees.
Thus, on converting $5\pi $ radians into degrees, we get 900 degrees.