Can the Area of a Sector be Determined Without Calculus or Trig?

[1MileCrash]

Homework Statement

Suppose we have a circle of radius r, and two points A and B on the circle.

We want to know the area of the sector cut off by A and B as a function of radius r and AB (the length of SEGMENT AB)

Without calculus or trig.

Homework Equations

The Attempt at a Solution

Part of the question included "is this possible" and after trying for a while, I think not.

Can I get a confirmation either way?

Thanks

 

[LCKurtz]

It depends on what is allowed "without calculus or trig", but I would say yes.

 

[1MileCrash]

My issue is that I feel like since the area of the sector involves the center angle - easily a transcendental given some interger AB value, that there isn't a way to get that transcendental without trig.

 

[LCKurtz]

Of course, you can argue that you don't even know the circumference formula without calculus, or even have the concept of "area" for a circle. That's why I say it depends on what's allowed. Also, I see that you are given the segment AB, which I took to mean the arc AB. If you mean the length of the chord, then I would be more inclined to say no.

 

[1MileCrash]

You're right, I should clarify.

I am allowed to use "high school geometry."

I know the area of a circle, I know Pythagorean theorem, I know area of a sector (in terms of radius and central angle) and so on.

But I could not say anything about the central angle with say, arcsin.

 

[1MileCrash]

We were trolled, no solution.

 

[HallsofIvy]

Who was "trolled" and by whom? Where did you get this problem? It's relatively simple to solve but does require trigonometry.

 

[1MileCrash]

My professor asked if it could be done, and if so, show how, for homework.

In class he revealed that it can't. So we were "trolled" in that we spent a lot of time on it because we figured that professors don't usually ask these if it really can't be done.