Radius of the circle.

Question

I am given that for two points $A, B$ on a circle, the length of the arc from $A$ to $B$ is $15$ whereas the length of the chord $\overline{AB}$ is $14$. How can I find the radius of the concerned circle?

Answer 1

If $r$ is the radius and $2\alpha$ the subtended angle, the the arc length is given by $15=2r\alpha$ and the chord length by $14=2r\sin\alpha$. Thus by eliminating $r$, we find the condition $$\tag1\frac{\sin \alpha}\alpha=\frac{14}{15} $$ for $\alpha$. Numerically$^{[1]}$, we obtain $\alpha\approx 0.63894677233$ and hence $r=\frac{15}{2\alpha}\approx 11.738$ and finally for the height of the segment $h=r(1-\cos\alpha)\approx 2.3156$.

$^{[1]}$ Unfortunately, there is no elementary way to solve $(1)$

Answer 2

$\theta=$angle subtended by half arc at the center $=\frac{15/2}{r}=\frac{15}{2r}\tag 1$

now, consider a right triangle $$\sin \theta=\frac{14/2}{r}=\frac 7r\tag 2$$ dividing (2) by (1), $$\frac{\sin\theta}{\theta}=\frac{7/r}{15/2r}=\frac{14}{15}$$ solving above equation gives, $\theta\approx 0.638947$

now, you can take from here