Running Around in Circles
If you have a pair of compasses near at hand, draw a circle of radius 10cm. Next use your compasses, stills set at radius 10cm, to divide your circle into 6 equal arcs. Now think about the following questions:
- Would using a radius of 20 cm divide your circle into 3 equal arcs? If not, what radius should you use?
- Would using a radius of 5cm divide your circle into 12 equal arcs? If not, what radius should you use?
A little experimentation with your compasses will produce approximate answers to these questions. (If you know some trigonometry you can calculate more accurate answers).
Number of arcs n | 3 | 4 | 5 | 6 |
Radius r (in cm) | 17 | 10 |
You can collect these and other answers in a table, then draw a graph to show the relationship between the number n of equal arcs and the radius r used. What will happen to the graph as the number of arcs increases?
For those of you who want to delve deeper, we can ask another question. Can we give a meaning to this problem of values of n less than 3?
One way of doing this is to define new variables x and y as follows:
| circumference of circle | ||
| x = |
| y = length of chord AB |
| length of arc AB |
,You will see that x and y are really just n and r in disguise. These new definitions allow us to consider values of x less than 3, including values that are not whole numbers. Do you agree with the extra values in the table below?
x | 0.75 | 1 | 1.2 | 2 | 3 |
y | 17 | 0 | 10 | 20 | 17 |
What will the corresponding graph look like? How does it behave for values of x less than 1? Can you use trigonometry to write down the equation of this graph? If so, try feeding the equation into the your graphics calculator or computer graph plotting package.
