I sometimes get bored at work and come up with math problems for myself. Out of necessity, I once figured up the arc length based on the chord length and radius. Just for fun, I came up with the area based on the chord length and radius. That wasn’t too hard, just figure out the area of the sector and subtract the triangular part. Yesterday I decided to tackle finding an equation for the area based on the lengths of the chord and arc, which is proving daunting even though I know there’s a solution.
I’ve tried listing all equations I can come up with for the unknown parts of the segment/sector, and substituting in equations with other knowns until I only have chord length and arc length in the final equation, but I just can’t get the unknowns out of the equation. I have a feeling there’s something simple I’m missing, but I can’t figure this out.
By the way, my knowledge of calculus is very limited, so if this can be done with just trigonometry and algebra (and I’m pretty sure it can), I’d prefer that method. And I’m interested in the steps in figuring out the equation, not just the equation itself.
If you can determine the arc length with the radius and the chord length, then can you not determine the radius with the arc length and the chord length? What am I missing?
It depends on what “determine” means. For example, consider a function like f(x) = x^5 + x + 1. Easy to figure out f(x) from x, right? Now, supposing I want to invert this to get x in terms of f(x); it’s not necessarily so easy to get an appreciably “closed-form” solution. [Depending on what “closed-form” means, it’s not necessarily even possible]
There’s not going to be any analytic solution for the radius because you have the angle both outside and inside a trigonometric function. You’ll have to resort to numerical methods.
As the above posters have said, there’s not a “closed-form” solution. The easiest way to find the answer for a given arc length a and chord length c is to find the radius r, which is given implicitly by
c/2r = sin(a/2r).
Once you’ve got a numerical value for r, it’s easy-peasy to figure out the area; but it’s not possible to write down a formula of the form r = foo*bar.
If you have the chord length and radius, then you have the opposite and hypotenuse of the triangular part of the sector which allows you to calculate the angle. From there, it’s pretty easy. The arc length has no direct relationship with that triangle.
I’m convinced there is some more complex way to figure it out though. For every ratio of the arc length to the chord length, there is one ratio of the radius to the chord length. For instance, when the arc length and chord length are equal, the radius is infinite and the area is zero. When the arc length is pi times the chord, the segment is a semicircle, the radius is half the chord length, and the area is half that of the whole circle. And there is a relationship between segments that are more than half the circle and less than half the circle. But I can’t figure out an equation that applies for all ratios even though I’m convinced there is one.
Just because there is a functional relation here doesn’t mean you’ll be able to specify that function in any nicer way than “The function which sends an arc length and chord length to the area of the corresponding segment”.
Look at it this way: if no one had ever bothered to name the trigonometric functions, what would you say if someone asked you how to figure out ratios of sides of triangles in terms of their angles? Instead of getting to say “Take the sine”, you’d have to say things like “Use the function which sends an angle to the corresponding ratio between the opposite side and the hypotenuse in a right triangle”. But it’s not as though the former is really any more informative than the latter. It’s just that we’ve already got a conventional name for it.
Well, now, you’re in the same situation, but without an existing conventional name. Alas.