Same Arc Length

If an arc of 45^\circ on circle A has the same length as an arc of 30^\circ on circle B, find the ratio of the area of circle A to the area of circle B.
Source: NCTM Mathematics Teacher


Let r_A be the radius of circle A and r_B the radius of circle B.
circunference\;of\;A=2\pi r_A
circumference\;of\;A=\displaystyle\frac{360^\circ}{45^\circ}\left (a\right )=8a
2\pi r_A=8a

circumferencce\;of\;B=2\pi r_B
circumference\;of\;B=\displaystyle\frac{360^\circ}{30^\circ}\left (a\right )=12a
2\pi r_B=12a

\displaystyle\frac{2\pi r_A}{2\pi r_B}=\frac{8a}{12a}

\displaystyle\frac{area\;of\;A}{area\;of\;B}=\frac{\pi r_A^2}{\pi r_B^2}
=\left (\displaystyle\frac{r_A}{r_B}\right )^2
=\left (\displaystyle\frac{2}{3}\right )^2

Answer: \displaystyle\frac{4}{9}


About mvtrinh

Retired high school math teacher.
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