circlesgeometry
I have the length of an arc of a circle
arc1 = 264cm
arc2 = 418cm
arc3 = 413cm
arc4 = 167cm
arc5 = 75cm
arc6 = 76.5cm
arc7 = 158.5cm
The lenghts are with a radius of 250cm.
I want to know how long these arcs become, when I expand the radius to 260cm.
What is the forumla to calculate this?
If $r$ is the radius then you have presumably found, as Isaac did, that $l=r\theta$ and $\frac{c}{2}=r \sin(\theta/2).$ You could eliminate $\theta$ to get $$2r \sin\left(\frac{l}{2r}\right) = c.$$ That you can solve for $r$ using numerical methods.
One way is to calculate the angle to the first point
$\alpha = \arctan \left ( \frac{p1.y-cp.y}{p1.x-cp.x} \right )$
Then you add your angle and calculate the new point:
$p2.x = cp.x + r * cos (\alpha +\theta )$
$p2.y = cp.y + r * sin (\alpha +\theta )$
Best Answer
Let the radius of the circle be $r$. Let an arc make angle $\theta$ (in radians) at the centre. Then, the length of the arc is $$l=r\theta$$ Here, you know $r$ and you know $l$. So, use that to find $\theta$. Then, take the same $\theta$, take your new radius, and use that to calculate the new $l$.
In other words, since $\theta$ is constant under "expansion of the circle", we must have $$\frac{l_1}{r_1}=\frac{l_2}{r_2}$$ So, $$l_2=\frac{l_1r_2}{r_1}$$ where $l_1$ and $r_1$ are the initial length of the arc and the radius respectively, and $l_2$ and $r_2$ are the final length of the arc and the radius respectively.