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Introducing Radians and pi

Did you know that, if you were to take a perfect circle, measure its radius (distance from centre to edge; for a perfect circle, this is the same all the way around), then measure the total distance around the edge (its circumference); with a piece of string perhaps, then you will find that the circumference is always approximately 6.28 times the radius, no matter how big the circle is?

Measuring angles in degrees

Of course you already know that there are 360 degrees in a circle. The degree is a very old unit of angular measurement (several millenia old, in fact). Ancient Sumerian people chose 360 because it's a number that divides cleanly into whole numbers, for example, 90 degrees for a quarter circle, 45 degrees for an eighth, etc. But the choice of 360 degrees is not fundamental in nature.

Measuring angles in radians

Nowadays however, all serious scientists and mathematicians measure angles in radians instead. There are approx. 6.28 radians in a full circle, meaning that, as stated above, the total distance around the edge is equal to approx. 6.28 times the circle's radius. That's all there is to it. It might help to think of the word 'radians' as a kind of plural of 'radius'.

But there's a problem. Measurements in radians may be natural and fundamental, but the numbers aren't clean or easy to work with. There are approx. 6.28 radians in a circle. That makes approx. 3.14 radians in half a circle. But that's only an approximate value. It's actually closer to 3.14159265, but as you'll come to appreciate in the wave mechanics section, even this value isn't exact enough for all applications. Consequently, this number has its own algebraic symbol: π, pronounced pi (It's actually a Greek letter). And there are people around who've somehow found the time to use their computers to calculate π to several million decimal places. The same miraculous number applies to all circles, and is fundamental to the way our universe works.

π is technically the ratio of the circumference (distance around) a circle, to its diameter (distance from one side of the circle to the opposite side).

So, there are π radians in half a circle; that makes π/2 radians in a quater circle, and 2π radians in a full circle. You get the idea. Quoting angles in radians in terms of π is just as easy; if not easier, than dealing with angles in degrees.

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