Calculator for finding the Area of a circle
   
 In algebra, and up to now in trig,
Related Chapters
   
 graphs have had an axis running left to right
 This axis is generally labeled X.
 There is also an axis running up and down.
 This axis is usually labeled Y.

 There is another way to draw a graph.
 This way is called a polar graph.
 It is set up to look something like a wheel.
 It has a center point.
 It has lines going out from the center point like spokes of a wheel.
   And it has rings around the center point.
 The whole thing actually winds up looking like a dart board.

 The center point is labeled zero.
 The rings are numbered by their distance from the center.
 In the picture above, the rings might be numbered 1,2,3 and 4
 going out from the center.

 

Realize that there is nothing magic about the numbers 1,2,3,and 4.

The rings might also be numbered 5,10,15, and 20. 

 
 The spokes are labeled with degrees,
 starting with the spoke at 3 O'clock and going counter clockwise.
 
 
 In an X Y graph, points are labeled with two numbers,
 one for the X axis and one for the Y axis.
 
 In a polar graph, a point is also labeled with two numbers.
 The first number is the distance the point is from the center.
 The second number is the degree line the point is on.
 
 So the point (2,30°) would be:

There is nothing really better or worse about polar graphing in general.
There are just times when it works better than other types of graphs.
Those times are usually when you are graphing equations that have trig functions.
 One trick you should know is how to convert something plotted on an X Y graph
 to a polar graph. 
 and back again.
 Let's look at the point (3,2) on an X Y graph.
 I bet you have made right triangles on X Y graphs
 to find the distance to a point before.
 We'll do that again here.

 
The point at the end of the hypotenuse (3,2) is our point.
The question is, how far is it from the origin (0,0) to our point (3,2).
  
To find the answer, just use the Pythagorean theorem:
 
 h2 = 22 + 32
h2 = 4 + 9
h2 = 13
h =
h = 3.606
 

 What is the measure of the little angle at the center of the graph? (Call it A)

 The tangent of an angle is equal to the length of the side opposite
 divided by the length of the side adjacent.
 So:
tan A = 2

3
 
tan -1 2 = A = 33.69°

3
 

 
  So for the X Y graph point: (3,2) the polar coordinates are: (3.606, 33.69°)
 
  Plotting that on the polar graph, we get:

 
  Now let's go from polar coordinates to X Y (called rectangular) coordinates.
 
 Find the X Y (rectangular) coordinates for the polar point (5,60°)

 
 We draw a right triangle on an X Y graph
 with a hypotenuse of 5 and a 60° angle on the left.
 

 
 Now we need to solve the triangle for X and Y.
 
sin 60° = Y

5
 
Y = 5 sin 60°
 
cos 60° = X

5
 

X = 5 cos 60°

 
 Let's stop right here for a moment.
 We wanted to find the X and Y coordinates of the polar point.
 

(5, 60°)

 And we found:

X = 5 cos 60°

and
Y = 5 sin 60°
 
 In general, for a polar point (a, B°) the X and Y coordinates are:
 
X = a cos B°
Y = a sin B°
 
 So let's solve it already!
 
X = 5 cos 60°
X = 5 (0.5)
X = 2.5
 
Y = 5 sin 60°
Y = 5 (0.8660)
Y = 4.33
 
 So the X Y coordinates of the polar point (5,60°) are (2.5,4.33)
 

copyright 2008 Bruce Kirkpatrick