Does the 1/360 of a circle have dimension?

Question

A circle has area$= \pi r^2$ , so the $1/360$ has area of $\pi r^2/360$ therefore 1degree is equals to $\pi r^2/360$ ?

Answer 1

I think you are a little confused. A degree is a measure of an angle (or, if you prefer, and are thinking in terms of radians, a measure of a fraction of the circumference of a circle). One degree is $\frac{2\pi}{360} = \frac{\pi}{180}$ radians, and is dimensionless.

Answer 2

The answer depends on the units in which you measure a "circle". $1/360$ of the area of a circle has units of area. $1/360$ of a $360$ degree angle has units of degrees. $1/360$ of the circumference of a circle has units of length. Radian angle measure is dimensionless since it's the ratio of arclength to radius, so $1/360$ of $2\pi$ radians is a number of radians, hence dimensionless.

Answer 3

A circle is not the same thing as $360^\circ$ and an angle is not the same thing as a wedge with an angular circlular arc spanning the angle.

So the "area of an angle" does not make sense.

However if you draw the following figure: two line segment each of length $r$; they share an endpoint; the angle between them is $n^\circ$; their other endpoints are connected via a circular arc as that the figure is a circular wedge; the circular arc will lie exactly on a circle that would be formed by a circle centered where the two line segments endpoints meet and the circle has radius $r$. That shape, a circular wedge, will indeed have an area of $\frac {\pi r^2 *n}{360}$.

But the circular wedge with an angle of $n^\circ$ is not the same thing as an angle of $n^{\circ}$.