Math/Trigonometry Question

[LaggyLou]

I am trying to figure out the length of an arc of a circle. I do not know the radius of the circle. I know the length of the chord that has the endpoints of the arc as its endpoints, and I also know the distance from the cord to the apex of the arc.

Intuitively I bvelieve I have enough information to get the length of the arc, but I am just at a loss [img]/images/graemlins/confused.gif[/img]. Van anyone help?

[PokerFink]

I'm thinking multiply the length of the cord by pi and then divide by 2?

[Evan]

This is wrong.

[jason_t]

[ QUOTE ]
I'm thinking multiply the length of the cord by pi and then divide by 2?

[/ QUOTE ]

This can't be correct as it doesn't use information about the distance of the chord to the apex of the arc. That is obviously important information.

[Evan]

R= (((A/2)^2) + (B^2)) / 2B
(R = radius) (A = Chord Length) (B = Arc Height)

Angle in Radians = 2 * Arcsin((Chord Length) / (2 * Arc Radius))

Arc Length = Arc Radius * Angle in Radians

[Jamey Saunders]

Don't take my word for this, because it's been 11 years since I had college trig, but...

If you know the distance from the chord to the apex (which, if I remember correctly is the center of the circle), and you know the length of the chord, then you know the lengths of the two legs of right triangles. The distance from the chord to the apex (let's call that X) and half the length of the chord (let's call the chord length Y) make up the sides of the triangle.

Knowing that, you could find the hypotenuse by solving C^2=X^2+(Y/2)^2. The hypotenuse would be the radius of the circle.

Where you go from here, I can't remember. But I do think I remember that there's a way to figure the opposing angle of a triangle if you know one angle (90*) and the lengths of a couple of sides. Once you've figured that angle, double it, and you've got the angle that includes the arc. Take that angle, and figure what percentage of the circumference it is. That percentage of the circumference of the circle should be the length of the arc.

That said, there's probably an easier way to do it. I'm waiting to see what others come up with.

[EDIT] Damn, Evan beat me to it, and he's much better than I am. Oh, well... [img]/images/graemlins/smirk.gif[/img]

[TBag]

Edit - btw x is chord height / 2.


I thought this worked, but Evan's is different.

When solving for radius, why do you divide by 2B?

[LaggyLou]

Thanks!!!!!!! OOT Rules, and I have now achieved the feat of billing my time for posting on 2+2!

Evan, if we ever meet, the drinks are on me!

[Evan]

[ QUOTE ]

Evan, if we ever meet, the drinks are on me!

[/ QUOTE ]
[img]/images/graemlins/grin.gif[/img]

[TBag]

Yargh, friggin trig. Hated that class. Anyways, could you explain to me why you divided by 2B when solving for the radius?

Isn't it just pythagorean's?