# Thread: area of a segment of a circle

1. ## area of a segment of a circle

Hello!

How do i get the area of this segment on the top of the circle?
w=44.5 mm
h=10.35 mm
r=34.9 mm

I have found some formulas for calculating the area, but it seems like I need some angle.. ? What If i dont know the angle...?
A=0.5*r*(b-w)+w*h
b is the arc length, b = r*angle*pi/180

How do I use this angle?
I know that the answer should be approx 696 mm2 from a computer program, but I cant do it analytically...

Need help

2. Erm , from your diagram , i guess W is the length of the chord . If so , we can apply this formula :

$\displaystyle 2r\sin\frac{\theta}{2}=44.5$ , We know the r and thus we can find $\displaystyle \theta$

So now you have the angle .

Area of a segment = $\displaystyle \frac{1}{2}r^2(\theta-\sin\theta)$

Erm , from your diagram , i guess W is the length of the chord . If so , we can apply this formula :

$\displaystyle 2r\sin\frac{\theta}{2}=44.5$ , We know the r and thus we can find $\displaystyle \theta$

So now you have the angle .

Area of a segment = $\displaystyle \frac{1}{2}r^2(\theta-\sin\theta)$

Thank you!

The expression 2*r*sin(theta/2)= 44.5. is that a general expression that always can be used or is it just for my case with my chord length ?

4. Originally Posted by danielel
Thank you!

The expression 2*r*sin(theta/2)= 44.5. is that a general expression that always can be used or is it just for my case with my chord length ?
With the chord w and two radii you have an isoceles triangle. Split it into two right triangles. (see attachment)

Then you get:

$\displaystyle \sin\left(\frac{\theta}2\right) = \dfrac{\frac12 \cdot w}{r}~\implies~ 2r\cdot \sin\left(\frac{\theta}2\right) = w$

5. Thank you!