How to Measure Angles

Date: 04/19/98 at 08:49:15
From: Sarah Smith 
Subject: Measuring Angles

Hi, 

I am having a lot of trouble with math right now because I do NOT 
understand how to measure angles at all. 

Thanks,
Sarah Smith

Date: 04/19/98 at 13:05:32
From: Doctor Luis
Subject: Re: Measuring Angles

First of all, don't panic! It's really easy to get lost in 
mathematical arguments if you don't visualize things the right way.

Anyway (to simplify things a bit), angles are a way to measure how
"wide" the separation between two intersecting lines is.

            /               |
           x                x
          /                 |   
         /                  |   
        /                   |   
       /                    | 
      / A                   | B
     .-----------x---       --------------x--

Looking at the two angles A and B above, you can intuitively say 
that angle B is wider than angle A, but how do you go about actually 
measuring the angles? Furthermore, you may also want to add angles or 
subtract them too, so you need some form of definition that will allow 
you to do arithmetic with angles.

If you consider two points (like the ones marked by the x's), as you
move further and further away (if you imagine sliding those points
away from the initial point of intersection, called the vertex of the
angle) you can see that the separation between those two points will
increase for both cases, BUT the distance will be greater for the
"bigger" angles. In the example above, if the distance from the vertex
to each x is the same for both cases, the distance between the x's is
greater for angle B than it is for angle A.

So you see, angles are a measure of that separation.

Now, how do we go about actually measuring those angles? Well
mathematicians have come up with a way to standardize the definition,
and what you do is consider a circle, but a very special sort of 
circle, one that has a radius of 1 (called the "unit circle")

          a
       x .  
     x     x          (we'll just call this a circle)
     x     . b          
       x x


If you take two points on the circle (like a and b) and draw a line 
from each of those points to the center of the circle you will get two 
intersecting lines. Now you can measure any angle with respect to your 
circle if you specify how wide the circle is. Some people like 360 
degrees, because 360 is a very nice number that is divisible by a lot 
of numbers like 12,10,5,6,8,3 and so on. There is another measure, 
which says that a circle is 2*pi radians, and this used more often in 
higher mathematics, because it simplifies some equations a little. If 
you don't know about it, "pi" is a very special constant that appears 
a lot in mathematics. It's approximately 3.14159.

Now, what you do is you compare your angle to the whole circle, and 
see what fraction of the circle you have. Remember we are measuring 
the angles with the center of the circle as the vertex.

For example, the angle B that we drew above corresponds to a quarter 
of the circle, and so we say that the angle measures 90 degrees (360/4 
since the whole circle is 360 degrees). If you were measuring the 
angles in radians angle B would be 2*pi/4 or pi/2 (in radians, the 
whole circle is 2*pi radians, so a quarter of that would be pi/2).

What if your angle corresponded to one eighth of the circle? Well, it 
would measure 360/8 or 45 degrees. And so on for other angles.

If you want to actually measure your angle *physically* there is an 
instrument called a protractor. It is basically a circle (or half of a 
circle), that has the 'circumference' (the edge) marked with numbers 
from zero to 360. You measure the angle by taking the protractor and 
placing the center of your circle exactly over the vertex of the angle 
you are trying to measure, then adjust it so that the zero goes over 
one side of the angle. Once you've done that, all you do is read the 
mark where the other side of the angle comes off (intersects) the 
circle (protractor) and that is the angle you are trying to measure.

So you see, the protractor works the same way as the unit circle.

It turns out that it doesn't matter how big your circle is, you will 
still get the same angle if you use a circle of any size. This is 
because the circumference (total length around the circle) scales by 
the same factor as the radius does, and so their relative size doesn't 
change if you make the circle bigger or smaller. Since the angle is a 
measure of that relative size, it doesn't change if you make the 
circle any size, and that is why we use circles to measure angles.

If you don't understand something, do not hesitate to reply and ask 
some more questions, and we'll be more than glad to help you. :)

Good luck.

-Doctor Luis,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/