Degrees and Radians, Explained

Date: 03/19/2002 at 23:09:20
From: Mandy
Subject: Degrees and radians

How do you find the degree measure for an angle from pi/60 rad?  I 
have a chart that I made of conversions, but I don't know how to do it 
when pi is added.  

If the radian measure of an angle is doubled, is the degree measure of 
the same angle also doubled?

Thank you.

Date: 03/20/2002 at 09:08:00
From: Doctor Peterson
Subject: Re: Degrees and radians

Hi, Mandy.

Radians and degrees are just different units for measuring angles, 
like feet and meters; so they are proportional, and when you double an 
angle, its measure doubles whether in degrees or radians.

Imagine drawing an angle, and then drawing a circle (the size doesn't 
matter) with its center at the vertex. We measure an angle by 
measuring the length of arc subtended by the angle - that is, by 
finding how long the part of the circle between the two sides of the 
angle is. (That's what a protractor does, for example; it's just a 
ruler curved around a circle.)

To measure in degrees, we might wrap a flexible measuring tape around 
the whole circle and mark off the circumference; then lay it out 
straight and divide that into 360 equal parts, so that each degree on 
the tape is 1/360 of the whole circumference. Now we can wrap it back 
around the circle, and measure the angle by counting degrees.

To measure in radians, we lay the tape straight and mark off the 
radius of the circle. That itself becomes our unit - do you see how 
much more natural that is than degrees? We can then divide the radian 
into whatever parts we choose, say hundredths. Then we wrap it around 
the circle and measure how many radians, in hundredths, the angle 
covers.

If you think about this, you realize that a whole circle will be 2 pi, 
or about 6.28, radians, since the circumference is 2 pi times the 
radius, which is the unit we used. Since a whole circle is also 360 
degrees, just because that is the number of parts we chose to divide 
the circle into, we see that

    360 degrees = 2 pi radians

Now we can convert between the two units by using the fraction

    360 degrees
    ------------ = 1
    2 pi radians

For example, given an angle of pi/60 rad, this is equal to

                360 deg    pi    360
    pi/60 rad * -------- = -- * ---- deg = 3 deg
                2 pi rad   60   2 pi

because we can cancel pi, 60, and 2 from the product.

It is also important to know that the area of a sector is proportional 
to the angle, so that for example a 90 degree sector is 90/360 = 1/4 
of a circle, and has area 1/4 pi r^2.

Write back if you need more help.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/