Degree or not degree, that is the question

by Andrew Irving (on Monday 15th September 2014)*

When someone decided that one 360th of a circle was so deeply significant a slice that it should contain precisely one degree, a unit was born!

By all accounts, this was an odd sort of choice. But the idea of a common language for angles was a good one and degrees weren’t all bad – they captured rational portions of circles. And that’s more than can be said for their peer, the radian.

Sector marked with cross captures an angle of 1 radian (lengths of radius and arc are equal) which is around 60 degrees.

Surprisingly though, radians make a lot more sense! Because unlike degrees, their size is not arbitrarily defined – for any circle’s sector,

$\mbox{angle in radians} = \frac{\mbox{arc length}}{\mbox{radius}}.$

Here, an angle is defined in a very natural way – in terms of the sector’s perimeter.  And this is why radians are Science’s unit of choice for angles.  But they also highlight something you may not know.

You see, when assessing a system’s behaviour, engineers use a technique called dimensional analysis to identify the contributing factors and how they relate to each other. This gives rise to a timeless principle – dimensions must balance. Now, units indicate dimension, e.g. with,

$\mbox{speed (m/s)} = \frac{\mbox{distance (m)}}{\mbox{time (s)}}$

we can see that the left hand side’s units are ‘metres over seconds’ (making its dimensions ‘length over time’) and the right hand side’s units are ‘metres’ over ‘seconds’ (making its dimensions ‘length’ over ‘time’).  We have balance.

As this example demonstrates, engineers measure physical properties in standard units (such as radians) that reflect their dimensions (e.g. length, mass, time, &c.). And this is what makes angles so curious – just look at our equation again,

$\mbox{angle (radians)} = \frac{\mbox{arc length (m)}}{\mbox{radius (m)}}$

and let’s carry out our own dimensional analysis! Well, both of the right hand properties are lengths – dimensions which cancel each other out. And dimensions must balance, meaning the left hand side has units but no dimension.  Which gives engineers one less thing to worry about!

So there it is: radians are both natural and convenient making angles measurable, yet dimensionless.

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* This blog was based on the YouTube video entitled ‘Rotary Motion’ by Sarah Peterson.