Need help putting a few parts of information together...

[gecko5068]

Ok if I have the arc length and rise of any given circle, shouldn't I be able to find the diameter of the circle? I'm studying this independently, so I don't have a book to rely on.

In other words, the arc length and rise ARE given, but I need to find the diameter given only those.

 

[galactus]

Yes, you can.

The formula for the arc length is \(\displaystyle \L\\S=R{\theta}\)...[1]

The formula for the 'rise', also known as the middle ordinate, is

\(\displaystyle \L\\M=R(1-cos(\frac{\theta}{2}))\)...[2]

Solve [1] for \(\displaystyle \L\\{\theta}\)

We get:

\(\displaystyle \L\\{\theta}=\frac{S}{R}\)

Sub into [2]:

\(\displaystyle \L\\M=R\left(1-cos(\frac{\frac{S}{R}}{2})\right)\)...[3]

Now, you know S and M, solve for R. Once you know R, the diameter is 2 times that.

Let's use an example. Say, M=5 and S=100.

From [3]: \(\displaystyle \L\\5=R\left(1-cos(\frac{100}{2R})\right)\)

We find that R=249.16, 9.59, 6.57, 3.02.

The diameter would be twice those.

498.32, 19.18, 13.14, 6.04.

The only one which works is R=249.162.

Therefore, D=498.32

100/249.16=0.401

That's about a 23 degrees angle subtended.

 

[gecko5068]

Thanks, now I'll get to work writing it in terms of R =)

 

[galactus]

It may prove difficult to solve that equation for R, generally. Enter in your known S and M, solve for R using technology.