$\pi$ as $180^{\circ}$ or $3.14$ in formula of areas. I know that may be it is a very simple question.
I came across through a question where they canceled $2π$ by $360^{\circ}$. Case was of the area of sector of a circle.
So I am not completely satisfied with the thing that we can change $\pi$ as $3.14$ or $180^{\circ}$ according to our need and I need a full explanation.
For clarity can $\frac{\theta\cdot \pi r^2}{360°}$ be written as $\frac{\theta\cdot r^2}{2}$ as per need.
 A: Yes. Radians can be converted to degrees. 
$\pi=180°$
So if you are given,
$$\frac{ \theta \pi r^2}{360°}$$
It is easier to convert everything to the same unit. We'll obtain,
$$\frac{ \theta \pi r^2}{2 \pi}=\frac{\theta r^2}{2}$$
My point is radians can be converted to degrees and it is easier to use one unit. Just like if you wanted to calculate the mean temperature it would be easier to use one unit (celsius or farhenheit).
A: I won't address the expression you brought up, just the first question.
$\pi$ radians is approximately $3.14$ radians. So, just approximating the constant $\pi$ as $3.14$ gives you the same kind of thing; a number of radians, as an angle measure.
On the other hand, it's completely accurate to say that $\pi \text{ radians } = 180^\circ$. Here, we're not approximating the number $\pi$, but switching to a new unit of angle measure; from radians, to degrees.
Here is an analogous situation, $1.60934$ kilometers could be approximated to $1.6$ kilometers; the same units, but less accuracy. We could also say that $1.60934$ kilometers "equals" $1$ mile. That time we've switched units.
