radians.
We do this like any other unit conversion. We want to change the units without changing the quantity. We do this by multiplying by 1, but the "1" by which we multiply should be a fraction with differernet units in the numerator and denominator.
ie, something like 
or 
. Both of these are equal to 1 since 12 inches is a foot and 60 minutes is an hour. Multiplying by "1" won't change our quantity, but it will allow us to convert units.
To convert between degrees and radians the "1" that we use will be either 
or 
.
Convert from degrees to radians:
Convert from radians to degrees:
centered at the point 
is the set of points 
that are distance 
from the point 
.
We can use the distance formula to find an equation describing these points.
so
We call this the equation of the circle.
The unit circle is the circle of radius 1 centered at the origin, the point 
. if we plug these numbers into the general equation of the circle we get the equation for the unit circle:
So points 
on the unit circle must always satisfy the equation 
.
with a central angle of 
is
Similarly, the area of the sector is
So for a circle of radius 
, one revolution corresponds to a linear distance of 
.
So if you pedal a unicycle at 2 revolutions per second and the unicyle has a wheel of diameter 18in, you would be traveling at a (linear) speed of:
We could convert this to mph:
Suppose you have two wheels, one of diameter 4 inches and one of radius 1.25 inches. The centers of the wheels are 6 inches apart. The larger wheel turns at a speed of 1000 rmp (revolutions per minute). At what speed does the smaller wheel turn?
Lets call the wheel with 4 inch diameter "wheel A".
Call the wheel with 1.25 inch radius "wheel B". The diamter of wheel B is 2.5 inches (diamter = 2× radius).
The "larger wheel" is wheel A.
Wheel A's angular speed is converted to the linear speed of the belt. Every time wheel A goes around one time, the belt travels by a linear distance equal to the circumference of wheel A= 4π inches.
So the speed of the belt is:
The speed of the belt gets converted back to the angular speed of wheel B. The circumference of wheel B is 2.5π inches, so every time the belt moves 2.5π inches wheel B will turn one revolution.
Speed of wheel B:
