Is it wrong to say that $\theta$ has to be given in "radians" (rather than "circular measure") in, e.g., the formula $\frac12\theta r^2$? Definitions:

A radian is the measure of the central angle subtended by an arc equal in length to the radius of a circle. The SI symbol for a radian is $rad$


The circular measure of an angle is the number of radians it contains.

Question body:
Consider the formula below which is used to calculate the area of a sector of a circle:
$$s = \frac{1}{2}\theta r^{2}$$
In many textbooks for introductory geometry, authors emphasize that the formula holds if $\theta$ is given in radians. However, when we compute, we use the circular measure of an angle given in radians. For example, let the radius of a circle be $6$ and the angle subtended at the centre be $\frac{\pi}{3}$ $rad$, hence we compute as follows:
$$s = (\frac{1}{2})(\frac{\pi}{3})(6^{2}) = 6\pi$$
As you can see, $rad$ is never included because we consider the circular measure of an angle, which is given in radians. Therefore, it yields $6\pi$ and not $6\pi$ $rad$.
Question:
As the title suggests, does $\theta$ represent the cirular measure of an angle or just an angle given in radians? Also, please explain why.
 A: Yes. The angle must be given in radians, or it can be the circular measure, or arc length, of the arc on the unit circle (a circle with radius 1). Think about a circle whose area is $s=\pi r^2$, and we want a portion of that area. Since a circle contains $2\pi$ radians, the area of the sector would be $s=\pi r^2\cdot\frac{\theta \textrm{ rad}}{2\pi\textrm{ rad}}$ and notice that it has to be radians here. Thus, $s=\frac{1}{2}\theta r^2$.
Also, a similar formula can be derived with degrees, but the denominator should be $360^\circ$ instead of $2\pi$. Thus, $s=\frac{\theta^\circ}{360^\circ}\pi r^2$, and remember that $\theta$ here is in radians.
A: $$\theta=180°=\pi\text{ rad}$$ just like $$l=1\text{ mi}=1.60934\text{ km}.$$
In principle, the unit should always be specified, but in practice it is implicit by context.
Then in formulas like
$$l=\theta r\text{ or }a=\frac{\theta r^2}2,$$ there is an implicit factor $1\text{ rad}^{-1}$, making them valid for $\theta$ in any unit.
A: Let me quote two passages from the question with emphasis added to a few key words.
Here is what the textbooks in question say:

... the formula holds if $\theta$ is given in radians.

And here is an explanation of the correct method according to you (which I agree with):

As you can see, rad is never included because we consider the circular measure of an angle, which is given in radians.

Is it possible the textbooks are saying the same thing as you? They even use some of the same words.
It seems to me that what the textbooks want you to do is to use circular measure. They have merely chosen not to use the particular words "circular measure" to describe it. Instead we have the words "given in radians", which in that context means the number $\theta$ is set to the number of radians that the angle contains,
so that $\theta$ is the circular measure of the angle.
A: Radians are defined as a ratio: the ratio of the subtended arclength to the radius.  For example, if the radius of the circle is 2 feet and the arclenth subtended by an angle is 4 feet than the angle $\theta$ measured in radians is $$\frac{4 \text{ft}}{2 \text{ft}} = 2$$ with no unit since the units cancel.  We call this 3/2 radians when it is useful, to remind us that we are working in radians; however the measure of an angle in radians is actually a unitless quantity. In other words, saying $\theta = 2 $ rad is the same as saying $\theta = 2$. $1\, \text{rad} = 1$.
If you want to you can also think of $\theta$ as an arclength along the unit circle; however this is also dimensionless because the unit circle has radius $1$, not radius $1$ foot or $1$ meter.
