What is the advantage of measuring an angle in radian(s)?
Look at the following picture. It is a circle of radius $r=1$ and there is an angle $\alpha$ that cuts out an arc $c$ from the circumference of the circle.
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How long is this arc? Well, this is where radians are immensely beneficial. If you express the angle in radians, then the length of the arc is exactly $\alpha$. If $\alpha=0.123$, then the arcs length is $0.123$ too. Easy.
You know that the full angle in radians is $2\pi$. If you choose $\alpha=2\pi$ in the above image, you would describe an arc $c$ that is actually just the full circle. And you immediately see, that its circumference (here also the length of the arc) is $2\pi$ too. This is exactly what the formula for the circumference $U(r)=2\pi r$ would give you anyway.
Of course this is also useful for circles of radii other than $1$. If your circle has a radius $r$ and you look at an arc cut out by an angle $\alpha$ in radians, then the length of the arc is $\alpha r$. It just scales linearly.
This is the visual aspect. But there are more (mathematical) reasons. Ask yourself, what would be a natural unit to measure an angle in? Degree is not very natural. There is nothing special about the number $360^°$ but its high divisibility. You could measure an angle in the interval $[0,1]$. Or you can use its natural connection to the circle that I explained above.
However, there are many mathematical functions that take angles as inputs, e.g., $\sin(x),\cos(x)$, etc. But have you ever asked how to actually compute $\cos(x)$ without a calculator? There are formulas that give very good approximations, e.g.,
$$\cos(x)\approx 1-\frac12 x^2+\frac1{24}x^4.$$
But they only work for radians. You can write them down for other units, but they will never look so natural, not even for angles in the units $[0,1]$.
With radians, the arc length and sector area are especially $r\theta$ and $\frac{1}{2}r^2\theta$. We also have trigonometric formulae $$\sin\theta=\sum_{n\ge 0}\frac{(-1)^n\theta^{2n+1}}{(2n+1)!},\cos\theta=\sum_{n\ge 0}\frac{(-\theta^2)^n}{(2n)!}.$$To work with degrees instead, you'd have to replace $\theta$ in each result with $\frac{\pi\theta} {180}$. Which would you rather use?
Degrees is the usual measure unit for angles, radians is the mathematical one. Degrees come from the historical base 60 operations. This base was chosen because it is divisible by the first six positive integers ($1$ to $6$). As it makes it easy to express common angles in degrees, its usage has persisted through centuries.
The radian on the other hand is a mathematical unit. An angle in radians is $ \frac {\operatorname{arc}} {\operatorname{chord}}$. You also have nice trigonometric functions with angles in radians: $\sin' = \cos$ and $\cos' = -\sin$.
That's the reason why we use degrees in daily life and radians in mathematics.