# Need help developing the formula to calculate the length of the y axis of a right triangle with a curved side for any position on the x axis.

If a right triangle has one side that is 500, another side that is 208, and the last side with a radius of 705, what is the formula to determine the length of the intersection point (y) at any given value of the x axis?

http://www.flickr.com/photos/trogdon/8048309563/in/photostream

-
What do you mean by "and the last side with a radius of 705"? is the curve shown part of a circle with radius = 705? This can't be because you show value 208 at the other end the O. So, the radius is either 208 or 705. Which one is correct? – NoChance Oct 2 '12 at 20:42

If the center of the circle is assumed vertically above the left vertex (which fits well as the difference between $(705-208)^2+500^2$ and $705^2$ is "negligibly small" $16$), then $$y=\sqrt{705^2-250^2}-\sqrt{705^2-500^2}\approx 161.$$