Blog: The areas of two circles have the ratio 49 : 64 . Find the ratio of their circumferences?

Tuesday, April 16, 2013

The areas of two circles have the ratio 49 : 64 . Find the ratio of their circumferences?

The area of a circle is $\displaystyle{A}=\pi{{r}}^{{2}}$ , and the circumference of a circle is $\displaystyle{C}=$ $\displaystyle{2}\pi{r}$ , where r is the radius of the circle. Applying this formulas to the two circles with radii $\displaystyle{r}_{{1}}$ and $\displaystyle{r}_{{2}}$ , we get

$\displaystyle{A}_{{1}}=\pi{{r}_{{1}}^{{2}}}$ and $\displaystyle{A}_{{2}}=\pi{{r}_{{2}}^{{2}}}$ .

Since the ratio of the two areas is 49 : 64, then

$\displaystyle\frac{{A}_{{1}}}{{A}_{{2}}}=\frac{{\pi{{r}_{{1}}^{{2}}}}}{{\pi{{r}_{{2}}^{{2}}}}}=\frac{{49}}{{64}}$

$\displaystyle\pi$ cancels from the numerator and denominator, and after taking square root we get the ratio of the two radii:

$\displaystyle\frac{{r}_{{1}}}{{r}_{{2}}}=\sqrt{{\frac{{49}}{{64}}}}=\frac{{7}}{{8}}$ .

Now we can find the ratio of the two circumferences, $\displaystyle{C}_{{1}}={2}\pi{r}_{{1}}$ and $\displaystyle{C}_{{2}}={2}\pi{r}_{{2}}$ :

$\displaystyle\frac{{C}_{{1}}}{{C}_{{2}}}=\frac{{{2}\pi{r}_{{1}}}}{{{2}\pi{r}_{{2}}}}=\frac{{r}_{{1}}}{{r}_{{2}}}=\frac{{7}}{{8}}$ .

The ratio of the circumferences of the two circles is 7:8.

Alternatively, this answer could be obtained without writing out the formulas by applying proportional reasoning. If the area of a circle is proportional to the radius squared, and circumference is proportional to the radius, then the ratio of the two circumferences must be the square root of the ratio of the two areas, which is

$\displaystyle\sqrt{{\frac{{49}}{{64}}}}=\frac{{7}}{{8}}$ .

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Tuesday, April 16, 2013

The areas of two circles have the ratio 49 : 64 . Find the ratio of their circumferences?

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