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Area of circle sector

I'm having trouble deriving the area of a circle sector formula, 12r2x\frac{1}{2} r^2 x. Any help?

I've got,

A=12r2cosxsinxA = \frac{1}{2} r^2 \cos x \sin x and B=π4r2sinx(1cosx) B = \frac{\pi}{4} r^2 \sin x (1 - \cos x).

The total area of the sector is A + B, and I can see my answer 'works' by letting x tend to zero... I'm just wondering why if I plug in a value of x to my A+B equation, I don't get what I know the answer should be.
Reply 1
Original post by Zuzuzu
I'm having trouble deriving the area of a circle sector formula, 12r2x\frac{1}{2} r^2 x. Any help?

I've got,

A=12r2cosxsinxA = \frac{1}{2} r^2 \cos x \sin x and B=π4r2sinx(1cosx) B = \frac{\pi}{4} r^2 \sin x (1 - \cos x).

The total area of the sector is A + B, and I can see my answer 'works' by letting x tend to zero... I'm just wondering why if I plug in a value of x to my A+B equation, I don't get what I know the answer should be.

Can't you just work out the fraction of a circle's total area the sector has, and then multiply that by the area of a circle?
Reply 2
Original post by Fallen
Can't you just work out the fraction of a circle's total area the sector has, and then multiply that by the area of a circle?


I could, but I want to know why my way isn't working. :tongue:

I haven't been set the question.
Reply 3
Original post by Zuzuzu
I could, but I want to know why my way isn't working. :tongue:

I haven't been set the question.

For starters, A isn't correct. What happens if x > pi/2 ?

(Or rather your formula for A is correct, but it is not a useful area to be trying to work out. Try to find a different triangle to calculate the area of).

Edit:
If you want to do this in a different way, might I suggest polar integration. This is a very odd thing to be doing (and B will be a bitch to calculate).
(edited 11 years ago)
Reply 4
Where did you get B from?
Reply 5
Original post by Fallen
For starters, A isn't correct. What happens if x > pi/2 ?

(Or rather your formula for A is correct, but it is not a useful area to be trying to work out. Try to find a different triangle to calculate the area of).

Edit:
If you want to do this in a different way, might I suggest polar integration. This is a very odd thing to be doing (and B will be a bitch to calculate).


I know how to get the formula in other ways. I'd just like to know why my answer doesn't work (for 0 < x < pi/2). If x = pi/3 the area of the sector should be pi/6 r^2, but A + B is something different.
Reply 6
Original post by TenOfThem
Where did you get B from?


Quarter of an ellipse with radii rsinx and r - rcosx.
Reply 7
Original post by Zuzuzu
Quarter of an ellipse with radii rsinx and r - rcosx.

It is not a quarter of an ellipse (unless x = Pi/2).
Reply 8
Original post by Fallen
It is not a quarter of an ellipse (unless x = Pi/2).


Ah, thanks.

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