# Coordinates of sector of circle

I know the coordinates of one point on a circle, this point is part of a sector. I know the angle of the sector at the centre of radius, I know the radius and I know the arc length. How do I calculate the coordinates of the other coordinate of the sector ? I have quickly mocked up the problem in the image above, please note it is B I am aiming to find.

Arc length = 5.5

A = (0.5,21)

• You can use complex numbers to calculate the co-ordinates of the new points . Take the init point as $0.5+21i$ and multiply it with $e^{i\theta}$ where $\theta =$ the radians by which it is rotated – abkds Mar 5 '14 at 12:20
• In contrast to your other question, this time you have too much information. The radis and angle tell you the arclength. – bubba Mar 5 '14 at 12:23
• The center is at distance $21$ from $(0.5,21),$ so it can't be $(0,0).$ On the other hand, if the distance is just very slightly greater than $21$ then $(0,0)$ can be the center. At least some of the numbers must be approximations since $21\times0.262 = 5.502$ (not $5.5$). – David K Dec 22 '18 at 5:44

Let $A = (x,y)$ and let $B= (x_1, y_1)$. Then $$x_1 = \; x\cos(0.262) + y\sin(0.262) \\ y_1 = -x\sin(0.262) + y\cos(0.262) \\$$
This assumes the center is again at $(0,0)$.