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Circles and Pi
Haagis
Introduction
Degrees and Radians
Tangents, Chords and Arcs
The Circle Theorems
Cyclic Polygons
Spheres, Cones and Cylinders
Conic Sections

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Circles and PiDegrees and Radians

Lugemise aeg: ~25 min

So far in geometry, we've always measured angles in degrees. A full circle rotation is °, a half circle is °, a quarter circle is °, and so on.

The number 360 is very convenient because it is divisible by so many other numbers: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, and so on. This means that many fractions of one circle are also whole numbers. But have you ever wondered where the number 360 comes from?

As it happens, 360 degrees are one of the oldest concepts in mathematics we still use today. They were developed in ancient Babylon, more than 5000 years ago!

At that time, one of the most important applications of mathematics was in astronomy. The sun determines the four seasons, which farmers have to know about when growing crops. Similarly, the moon determines the tides, which was important for fishers. People also studied the stars to predict the future, or to communicate with gods.

A Babylonian tablet for calculating 2

Astronomers noticed that the constellations visible at a specific time during the night shifted a tiny bit every day – until, after approximately 360 days, they had rotated back to their starting point. And this might have been the reason why they divided the circle into 360 degrees.

Midnight on day ${day}

Of course, there are actually 365 days in one year (well, 365.242199 to be exact), but Babylonian mathematicians worked with simple sundials, and this approximation was perfectly adequate.

It also worked well with their existing base-60 number system (since 6×60=360). This system is the reason why we still have 60 seconds in a minute and 60 minutes in an hour – even though most other units are measured in base 10 (e.g. 10 years in a decade, or 100 years in a century).

For many of us, measuring angles in degrees is second nature: there is 360° video, skateboarders can pull 540s, and someone changing their decision might make a 180° turn.

But from a mathematical point of view, the choice of 360 is completely arbitrary. If we were living on Mars, a circle might have 670°, and a year on Jupiter even has 10,475 days.

The 540 McFlip, a 540° rotation

Radians

Rather than dividing a circle into some number of segments (like 360 degrees), mathematicians often prefer to measure angles using the circumference of a unit circle (a circle with radius 1).

A has circumference .

For a , the corresponding distance along the circumference is .

For a , the distance along the circumference is .

And so on: this way of measuring angles is called radians (you could remember this as “radius units”).

Every angle in degrees has an equivalent size in radians. Converting between the two is very easy – just like you can convert between other units like meters and kilometers, or Celsius and Fahrenheit:

360° = 2π rad

⇒
1° = rad

⇒
1 rad = °

You can write the radians value either as a multiple of π, or as just a single decimal number. Can you fill in this table of equivalent angle sizes in degrees and radians?

degrees060180
radians0232π

Distance Travelled

You can think of radians as the “distance traveled” along the circumference of a unit circle. This is particularly useful when working with objects that are moving on a circular path.

For example, the International Space Station orbits Earth once every 1.5 hours. This means its speed of rotation is radians per hour.

In a unit circle, the speed of rotation is the same as the actual speed, because the length of the circumference is the same as one full rotation in radians (both are 2π).

The radius of the ISS orbit is 6800 km, which means that the actual speed of the ISS has to be = 28483 km per hour.

${round(p*1.5,1)}h

Can you see that, in this example, radians are a much more convenient unit than degrees? Once we know the speed of rotation, we simply have to multiply by the radius to get the actual speed.

Here is another example: your car has wheels with radius 0.25 m. If you’re driving at a speed of 20 m/s, the wheels of your car rotate at radians per second (or 802π=13 rotations per second).

Trigonometry

For most simple geometry problems, degrees and radians are completely interchangeable – you can either pick which one you prefer, or a question might tell you which unit to give your answer in. However, once you study more advanced trigonometry or calculus, it turns out that radians are much more convenient than degrees.

Most calculators have a special button to switch between degrees and radians. Trigonometric functions like sin, cos and tan take angles as input, and their inverse functions arcsin, arccos and arctan return angles as output. The current calculator setting determines which units are used for these angles.

Try using this calculator to calculate that

sin(30°) = cos(1°) =
sin(30 rad) = cos(1 rad) =

DEG
7
8
9
sin
4
5
6
cos
1
2
3
tan
0
.
C
mode

Using radians has one particularly interesting advantage when using the Sine function. If θ is a very small angle (less than 20° or 0.3 rad), then sinθ≈θ. For example,

sin(${x}) ≈ ${sin(x)}…

This is called the small angle approximation, and it can greatly simplify certain equations containing trigonometric functions. You’ll learn much more about this in the future.

Archie