Is there a general formula for a “sector” of a circle - by “sector,” I don’t mean a sector defined by two radii and the arc of the circle intercepted by the radii. What I mean is:
Take any point in the circle. Now draw two line segments from that point to the circle, creating an arc. The two segments don’t have to be the same length (as they are in a traditional sector).
I’m curious if there’s a general formula for that area, which depends, I suppose, on:
the location of the point common to the 2 segments (this could be defined simply by the distance from the circle’s center)
the “central” angle (not really central, but the same concept)
Your statement doesn’t sufficiently define the situation. From any random point inside a circle I can draw an infinite number of lines from that point to the circle. You have to somehow define the two line segments
In other words, it’s like you’re cutting a piece of pie, with the “point” not at the center of the pie. Right? As **CalMeacham **said, we need more information.
Well, you can calculate areas by integration, and it’s possibly to analytically integrate a circle, so you could always convert the points to Cartesian coordinates (x,y) and figure out the solution through integration. That would be kind of a brute force method, there might be quicker ways.
This is needlessly complicated. The answer is quite simply the area of a triangle + the area of a circular segment. You just need to define the two from the starting information.
They gave more information… they gave not just 1), but also 2) and 3).
Anyway, here’s how I’d frame this problem:
Take a circular pie (of radius r, with center O), and pick point P inside the circle and points X and Y on the circle, making straight cuts of our pie from P to X and from P to Y. This cuts our pie into two pieces, and we want to know the area of the piece not containing O (the area of the other piece will, of course, be πr[sup]2[/sup] minus this).
Let A be the angle POX and let B be the angle POY, both measured in radians. Also, let d be the distance of P from O. If I’m not mistaken, the area we’re interested in is r/2 * [r(A - B) - d(sin(A) - sin(B))].
Explanation: In the case where one of the straight-line cuts lies along a radius of the circle, the area we’re interested in is given by an ordinary central circular wedge minus a triangle. Thus, r/2 * r * θ - r/2 * d * sin(θ), where θ is the relevant angle in radians.
And more generally, fixing the circle and P, but letting A and B vary, if f(A, B) is the corresponding area [oriented and counted with multiplicity as A or B wrap around the circle multiple times in any direction], then this takes the form g(A) - g(B) for some function g, allowing us to reduce this to the problem of the previous paragraph.
I said the formula would / could depend on the lengths of the two line segments. From a point that is not the center of the circle, there are at most 2 line segments with a given length - the given length defines the radius of a new circle centered at that point and two circles can intersect at most 2 times.
I suppose you’re right in the sense that I would need to define which of the 2 line segments I’m talking about, but I also said the formula would / could depend on the “central” angle, which should pinpoint which of the (at most) 2 line segments I’m drawing.
Your framing is more precise than mine and seems to capture what I had in mind.
I understand the first paragraph of your explanation (when one of the line segments lies along a radius of the circle) but I guess I’m not following exactly your next statement. I can see how if PX and PY lie on opposite sides of the radius OP, you can break the problem down into 2 instances of your first paragraph and actually ADD the results, but I don’t think that’s what you’re saying.
So for me, the “unsolved” case (or perhaps more accurately, “un-understood” case) is when PX and PY each lie on the SAME side of radius OP.
(BTW, I’m interested in this as I have a math background; I’m not a student and this isn’t HW, in case I need to make that clear.)