Area of A Segmented Circle Cut By A Chord

[TAMUwbEE]

Homework Statement

Given a circle with a set diameter, how does one calculate the area of the segment below?

Homework Equations

The only information available is the diameter (in this particular example it is 14"), You may not use angles. Only the chord length.

Thank you for your reply.
Miguel

 

[andrewkirk]

The area T of the triangle made by the circle centre and the chord can be expressed in terms of L (chord length) and R without using any angles.

But the area of the sector comprising that triangle and the segment is proportional to the angle subtended at the centre by the chord. So it seems to me we cannot calculate the area of that sector without using either that angle or something equivalent to an angle (eg an inverse trig function of a ratio of R- and L-related items).

If we can't calculate the sector area with those restrictions then we can't calculate the segment area either, since the difference between the two is just T.

 

[TAMUwbEE]

The area T of the triangle made by the circle centre and the chord can be expressed in terms of L (chord length) and R without using any angles.

But the area of the sector comprising that triangle and the segment is proportional to the angle subtended at the centre by the chord. So it seems to me we cannot calculate the area of that sector without using either that angle or something equivalent to an angle (eg an inverse trig function of a ratio of R- and L-related items).

If we can't calculate the sector area with those restrictions then we can't calculate the segment area either, since the difference between the two is just T.

Yeah that's the problem I'm running into - without the angle, it seems impossible. However, is there a solution possible using the sagitta.

 

[HallsofIvy]

If you can't do it without knowing the angle, then find the angle! If you draw a line from the center of the circle to the center of the chord, you have a right triangle with the length of the hypotenuse equal to the radius of the circle and "opposite side" equal to half the length of the chord: $sin(\theta)= L/R$. And the angle between the two radii is twice that.

 

[RUber]

You can start by thinking of how a function might look.
You know that if chord length C is equal to diameter D, then the area is that of half the circle. $\frac{\pi}{8}D^2$ and if C is zero, the area is zero. So you need a function of C that is defined on [0,D] with range on $\left[0,\frac{\pi}{8}D^2\right]$.

As HallsofIvy mentioned, the ratio of C/D is equal to the sine of half the angle connecting the ends of the chord to the center. So the most natural solution would be one involving an inverse sine function to find the angle, taking the appropriate portion of the total area of the circle, then subtracting off the area of the triangle with base length C and height found by applying the Pythagorean theorem.

You say that you have a solution. Perhaps you could post the solution so we can better understand the method being suggested.

 

[andrewkirk]

My understanding of the problem is that the approaches suggested in the last two posts are not allowed. That is, the solution is not allowed to calculate the angle along the way, either explicitly or implicitly ('You may not use angles').

Without that restriction, the problem is trivially simple.

I don't think using the sagitta can get around this restriction either, as the sagitta is a function of the cosine of the angle, so one has to do an inverse cosine operation to get something useable.

I'd be interested to know where this problem comes from, as it seems a very strange (and probably impossible) question.

 

[RUber]

https://www.physicsforums.com/threads/area-of-a-sector-from-a-chord.745192/

 

[andrewkirk]

Good find, RUber! Here's the conclusion of that linked thread:

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.

I think the question from the professor in the linked thread is fair enough - and not trolling, as he asked if it could be done, thereby hinting at the possibility that it couldn't. If the question asked in this instance was worded exactly as in the OP, then I think that is indeed an inappropriate question from the teacher, as it is asked in a way that implies it can be done, which is misleading. But perhaps it has been inaccurately transcribed.