# Section of a circle

I need some help with some of my homework, I can't figure it out.

I have a radius on a circle and a height from the circle to the chord.

I found this formula $$h=r \left(1-\cos \frac{v}{2} \right)$$ And isolated it to $$v = \arccos \left( \frac{h/r -1}{2}\right)$$ Not sure if that's correct. Then I input it in this formula $$A=r^2((\pi v)/360-(\sin v)/2)$$ I have tried with a radius of 1000 and a height from the circle to the chord of 1000.

But it gives a wrong result.

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What is given, what do you search for? So given: radius r, and distance to (some) chord h. What is $v$? – Simon Markett Jun 28 '12 at 11:57
I want to find the areal, of the circle section. R=1000, Distinace to chord=1000. Then i calculate v in the first formula. – NikolajSvendsen Jun 28 '12 at 12:07
When the radius and distance to the chord is exactly the same, it should be the half of the circle. But it gives a wrong result, i am not sure whetever i mixup radians and degress, or i isolated it wrong. – NikolajSvendsen Jun 28 '12 at 12:29

The problem is that your solution is incorrect. You have the 2 factor in the wrong place. the correct solution is:

                    v=2arccos((1-h/r)

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Thanks that works. But can you explain me how v, can be 180 degress. Isn't it a triangle like on this image matb1htx.systime.dk/fileadmin/indhold/ISBNXXXXXXXXXXXXX/… – NikolajSvendsen Jun 28 '12 at 12:59
In your example, the height to the chord is equal to the radius. Therefore the chord becomes a diameter and the angle subtended by a diameter is 180 degrees. – Barry Jun 28 '12 at 13:12

Yep, that is correct:
$v=2\arccos(1-\frac{h}{r})$

And this is how we get it:

$\cos(\frac{v}{2}) = \frac{r-h}{r}$
$\cos(\frac{v}{2}) = 1-\frac{h}{r}$
$\frac{v}{2} = \arccos(1-\frac{h}{r})$
$v=2\arccos(1-\frac{h}{r})$

See this homework help resource for more.

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