The length of an arc of a circle is $12$ cm. The corresponding sector area is $108$ cm$^2$. Find the radius of the circle.
I have not attempted this question and do not understand how to solve this.
[Math] finding radius from sector area and arc length of a circle
arc length
Related Solutions
Hint. Recall that the length of an arc of a circle and the corresponding angle subtended at the centre are proportional.
There is no closed-form formula.
Let $\theta$ be the half aperture angle. Then the half arc length is
$$a:=r\theta$$
and the half arc "width"
$$w:=r\sin\theta.$$
Taking the ratio, you get a transcendental equation
$$\frac{\sin\theta}\theta=\text{sinc }\theta=\frac wa$$ and $$\theta=\text{sinc}^{-1}\dfrac wa$$ (the inverse cardinal sine function), but this is not considered a usual function.
For the same reason, there is no closed-form formula for $r$, because that would create one for $\theta$.
For small angles, you can use the Taylor development to the fourth or even sixth degree, and solve the biquadratic or triquadratic
$$\frac wa=\color{#0247fe}{\text{sinc }\theta}\approx\color{#32cd32}{1-\frac{\theta^2}6+\frac{\theta^4}{120}}\approx\color{magenta}{1-\frac{\theta^2}6+\frac{\theta^4}{120}-\frac{\theta^6}{5040}}$$ then refine with Newton.
Best Answer
If $\theta$ is the central angle in radians and $r$ is the radius, we can write the two following equations: $r\theta = 12$ and $\frac{1}{2}\theta r^{2} = 108.$ Dividing the first equation into the second, we have that $\frac{1}{2}r = 9,$ so $\boxed{r = 18}.$