When an angle is drawn, it is not exactly as measured, as a negligible part of that angle is covered up by the lines making the angle. If we take the thickness of the line and measure it using angular measurements(for instance degree), could we use the same for expressing lengths using a combination of length and angle? (x cm as 5 degrees (from a distance y cm from a fixed point) But if in a measuring instrument(say protractor), 1 degree = 1mm (or any other unit), that would be the same as measuring 5 cm as 50mm. So can angles and lengths be considered the different sides of the same coin. Are angles and lengths equivalent
This seems awfully similar to the relation l=rθ.
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I'm not sure if I understand your question fully; it's unclear what you mean when you say "measure the thickness of the line using angular measurements."
As you point out, $\ell = r \theta$ measures the length of a circular arc that rotates $\theta$ radians and has a radius $r$. However, note that although radians are a measurement, they are in fact unitless! It sounds crazy, but it actually comes from the definition:
This means they are defined in terms of (length / length) and are have no unit. This is in contrast to lengths, which do. Put another way, the above equation is the same as saying that, by definition, the radians of an arc are given by $\theta = \ell / r$.
So, no, angles and lengths are not equivalent. In the specific case that you have a circular arc that traverses some angle, you can use the radius of the circle to relate it to a length.