```Question 255263
The length of an arc is, in general:
<pre>
degrees in the arc
------------------ * 2 * pi * r
360
</pre>
If {{{r[A]}}} represents the radius of circle A then the arc length is:
{{{(45/360)*2*pi*r[A]}}}
which simplifies to
{{{(1/4)pi*r[A]}}}<br>
If {{{r[B]}}} represents the radius of circle B then the arc length is:
{{{(30/360)*2*pi*r[B]}}}
which simplifies to
{{{(1/6)pi*r[B]}}}<br>
We are told that these arc lengths are equal so
{{{(1/4)pi*r[A] = (1/6)pi*r[B]}}}<br>
We are asked to find the ratio of the areas of the two circles. The area for circle A would be {{{pi*r[A]^2}}} and for circle B it would be {{{pi*r[B]^2}}}. The ratio then would be:
{{{(pi*r[A]^2)/(pi*r[B]^2)}}}
The pi's cancel leaving:
{{{r[A]^2/r[B]^2}}}
which can be rewritten as
{{{(r[A]/r[B])^2}}}
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
{{{(1/4)pi*r[A] = (1/6)pi*r[B]}}}
We can use this to find the ratio of the radii. Multiplying both sides by 4 we get:
{{{pi*r[A] = (4/6)pi*r[B]}}}
which simplifies to:
{{{pi*r[A] = (2/3)pi*r[B]}}}
Dividing both sides by pi we get:
{{{r[A] = (2/3)r[B]}}}
Dividing both sides by {{{r[B]}}} we get:
{{{r[A]/r[B] = 2/3}}}
So the ratio of the radii is 2/3. We can put this into our area ratio expression:
{{{(r[A]/r[B])^2}}}
and we get:
{{{(2/3)^2}}}
which simplifies to:
{{{4/9}}}<br>
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!?

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