```A battle for the ages…





Where Do Degrees Come From?
Before numbers and language we had the
stars.
Ancient civilizations used astronomy to
mark the seasons, predict the future, and
appease the gods (when making human
sacrifices, they’d better be on time).
How is this relevant to angles? Well, bub,
riddle me this: isn’t it strange that a circle
has 360 degrees and a year has 365
days?. And isn’t it weird that
constellations just happen to circle the
sky during the course of a year?
Unlike a pirate, I bet you can’t determine
the seasons by the night sky.
Degrees measure angles by how far we tilted our heads.
Constellations/stars make a circle every day. If you look into the night sky at the
same time every day (say midnight), the stars will make a circle throughout the
year. One theory about how degrees came to pass is:
•
•
•
Humans noticed that constellations moved in a full circle every year.
Every day, they were off by a tiny bit (“a degree”)
Since a year has about 360 days, a circle had 360 degrees
But, but… why not 365 degrees in a circle?
Cut ‘em some slack: they had sundials and didn’t know a year should have a
convenient 365.242199 degrees like you do…Remember about leap year!
360 is close enough for government work. It fits nicely into the Babylonian
sexagesimal base-60 number system, and divides well (by 2, 3, 4, 6, 10, 12, 15,
30, 45, 90… you get the idea).
Basing Mathematics on the Sun Seems Perfectly Reasonable
• Earth lucked out: ~360 is a great number of days to have in a year.
• On Mars we’d have roughly ~680 degrees in a circle, for the longer
Martian year.
• In parts of Europe they’ve used gradians, where you divide a circle
into 400 pieces.
• Many explanations stop here saying, “Well, the degree is arbitrary
but we need to pick some number.” Not here: we’ll see that the
entire premise of the degree is backwards.

Degrees measure angles by how far we turn our head.

Radians measure angles by the distance traveled.
Arc Length s

r
Radian        

s
r

A circle has 2 radians or 360 degrees.

Radians are a ratio and are written without units.

It helps to think of radians as the distance traveled on
the unit circle.







Radian or “rad” is the ratio between the length of an arc and its
The radian is the standard unit of angular measure, used in
many areas of mathematics.
One radian is approximately 57.295 779 degrees.
Using radians to measure angles seems unnatural at first.
However, when angles are stated in radians the constant π
tends to disappear from the equations, and this greatly
simplifies calculation.
For example, the length of an arc is simply its radius multiplied
by its angular measure in radians, and the area of a sector of a
circle is simply its angular measure in radians multiplied by half
The radian was defined and named by James Thomson in 1873.
Some suggest it may have been intended as an abbreviation for
So What’s the Point?
• Degrees have their place: In our own lives, WE are the focal point and
want to see how things affect US. How much do I tilt my head, spin my
snowboard, or turn my steering wheel?
• When we begin to study the world around us (with mathematics – called
science), we are an observer describing the motion of others. Radians are
• Even angles can be seen from more than one viewpoint, and
understanding radians makes math and physics equations more intuitive.
Happy math.
```