# Arcs of a circle Geometry and angles

How do I find out what the radius length of the angle is? The answer is D by the way.

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Let the circle's circumference be given by $C$, let it's diameter = $d$, it's radius $r$.

The ratio of the arc of the circle to the circumference is equal to the ratio of the $30$ degree angle to the measure of a circle = $360^\circ$:

$$\dfrac{6}{C} = \dfrac{30}{360} = \dfrac{1}{12}\quad \implies \quad C = 6\times 12 \implies C=72$$

Using the formula for circumference of a circle: $C = d\pi = 2\pi r$, and solving for r:

$$C = 72 = 2 \pi r$$ $$\implies r = \dfrac{72}{2\pi} = \dfrac{36}{\pi}$$

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is this off by a factor of 10? I think you meant 72 after reductions. Regards – Amzoti Feb 15 '13 at 21:38
Yes, indeed!... – amWhy Feb 15 '13 at 21:43
Is this making sense, Little Jon? – amWhy Feb 15 '13 at 21:48
Yes this is a valid answer Thank you! – Little Jon Feb 15 '13 at 21:52
You're welcome! – amWhy Feb 15 '13 at 21:53

The full circle has perimeter $\frac{360^\circ}{30^\circ}\cdot 6\,\mathrm m=72\,\mathrm m$. As this must equal $2\pi r$, we find $r=\frac{36}\pi\,\mathrm m$.

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HINT: The circumference of a circle of radius $r$ is $\pi r$. The $30^\circ$ angle is $\frac{30}{360}=\frac1{12}$ of the total angle at the centre of the circle, so $6$ metres is $\frac1{12}$ of the circumference of the circle. The whole circumference is therefore $6\cdot12=72$ metres, which, as already noted, is $2\pi r$. Therefore $r$ is ... ?

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Length of the arc is $L_\alpha = \alpha r$, so $r = \frac {L_\alpha}\alpha = \frac 6{\pi/6} = \frac {36}\pi$

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The way that you can solve this is by using the following formula:

$$A=\frac{n}{360}\pi{r^2}$$

Hope that helps you out! I can't figure out how to type in the equation!

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