How to the Find the Radius of a Sector

Question

I know how you find out the Area of a Sector and the Arc Length but I'm not sure how to find out the radius of a circle?

I understand that there are formulas but I find them quite confusing. Please can you explain the answer to this for me.

Thanks.

Answer 1

Using the formula $$L=2\pi r\times\frac{\theta}{360}$$ where $L$ is arc length, you can rearrange to give: $$r=\frac{L}{2\pi} \times \frac{360}{\theta}$$ You know that $L=15$ and that $\theta = 27$, so put these values into the formula to obtain: $$r=\frac{15}{2\pi}\times\frac{360}{27}=31.83cm$$ If you find this formula confusing I would think about it like this. The circumference of a whole circle is $\pi d = 2\pi r$. To find the arc length, we need to its percentage of the total circumference of the circle. This percentage is the angle in the middle of the sector divided by the angle of a whole circle ($\frac{\theta}{360}$). Hence to find the arc length, you find the circumference and then multiply this by the percentage of that circumference which we need to find. This is where the first formula comes from.

Answer 2

Just complete the whole circle, if the angle $2\pi$ of the whole circle means an arc length = $2\pi r$ so for an angle $27$ giving you an arc length $15$ you can deduce that $2\pi r(27)$ = $2\pi15$, hence $r = \frac{15}{17}$.

Answer 3

@PerfectNutter That is just an equation solving for r (radius): $15=2\pi r(\frac{27}{360})$. Step 1: Combine like terms. $15=0.471r$. Step 2: Divide each side by $0.471$ rounded to nearest hundredth. $31.85=r$. Step 3: Check your answer. $15=2\pi*31.85*(\frac{27}{360})$, $15=15.00$.