The length of an arc is, in general:
degrees in the arc
------------------ * 2 * pi * r
360
If represents the radius of circle A then the arc length is:
which simplifies to
If represents the radius of circle B then the arc length is:
which simplifies to
We are told that these arc lengths are equal so
We are asked to find the ratio of the areas of the two circles. The area for circle A would be and for circle B it would be . The ratio then would be:
The pi's cancel leaving:
which can be rewritten as
So if we can find the ratio of the radii then we can find the ratio of the areas. Our arc length equation tells us that
We can use this to find the ratio of the radii. Multiplying both sides by 4 we get:
which simplifies to:
Dividing both sides by pi we get:
Dividing both sides by we get:
So the ratio of the radii is 2/3. We can put this into our area ratio expression:
and we get:
which simplifies to:
Note that we were able to find the ratios of the radii and the areas without being able to find any radius or any area!?