Turn, not length
What Is an Angle Actually Measuring?
An angle measures how far something turns, which is why the arms' length does not matter.
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Step 1 of 5
An angle lives at the corner, not along the arms.
Two lines meeting at a point make an angle, and what is being measured is the amount of turn from one line round to the other.
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Step 2 of 5
Longer arms do not make a bigger angle.
Extending both lines makes the drawing larger but changes nothing about the turn, so the angle stays exactly the same size.
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Step 3 of 5
A full turn was split into 360 parts.
Degrees are an old choice, not a law of nature. 360 was picked because it divides neatly by so many numbers, which makes fractions of a turn tidy.
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Step 4 of 5
The familiar angles are just fractions of a turn.
A right angle is a quarter of the way round, a straight line is halfway, and anything past halfway starts folding back on itself.
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Step 5 of 5
Angles are how directions get compared.
A compass bearing, a ramp's steepness and a roof's pitch are all angles, because each is really a question about how far something has turned.
360 was probably chosen by Babylonian astronomers because a year is roughly 360 days and 360 divides by 24 different numbers.
The short version
An angle measures how far something turns between two directions, which is why stretching the arms longer does not change it.
Try it yourself
Open a door slightly and then fully. The angle changes, but sliding your hand along the edge of the door does not change it at all.
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