# get length of line connecting sector of a circle

What is the formula for getting the length of a line (in this case the red one) connecting starting point and end point of an arc, given the circle's radius $R$ and $\angle A$? • $2R \sin \dfrac{A}{2}$. – Anurag A Jun 16 '15 at 23:58
• @AnuragA thanks – tjvg1991 Jun 17 '15 at 0:08

Never mind. I got the answer to my own question. I used cosine law for this one.

Red Line length = $\sqrt{R^2 + R^2 - 2RRcos(A)}$

The triangle obtained by joining the end-points of the chord (red line) to the center of circle, we get an isosceles triangle. Thus, dropping perpendicular to the chord, we get right triangle in which we have $$\sin\frac{A}{2}=\frac{\left(\frac{\text{length of chord}}{2}\right)}{\text{radius}}=\frac{\left(\text{length of chord}\right)}{2R}$$ $$\implies \color{blue}{\text{length of chord}=2R\sin \frac{A}{2}}$$

As seen in the image below the length of the chord is $2R\sin(A/2)$.

The identity $\sin(A/2)=\sqrt{\frac{1-\cos(A)}2}$ demonstrates that tjvg1991's and Harish Chandra Rajpoot's answers agree: $$2R\sin(A/2)=\sqrt{2R^2-2R^2\cos(A)}$$

$\cos A = \frac{c^2 + b^2 - a^2}{2cb}$ can be used, where $A$ is the interior angle between sides $b$ and $c$.