Math problem (geometry)

[Andy Perreault]

A buddy at work asked me if I knew how to solve a geometry problem for him. I'll try to explain it as best as I can, hope you can picture what I mean. He's building a mantle above his fireplace, and he wants the edge to have a smooth arc. So, he needs to find out what the radius of the arc should be. When you draw a line anywhere through a circle except its center, it's called a chord. Okay, so he drew a circle with a line passing though it and said that the chord is to be 4 and a half feet long, or 54 inches. Now, the maximum thickness of the segment of the circle from the chord line to the edge of the circle is to be 1 and 1/2 inches. Picture drawing a happy face and putting a transparent hat across the top of his head. The brim of the hat is the chord line that is 54" long, and the distance from the brim of the hat to the very top of his head is 1. 5 inches. That's a pretty fat head! So, the answer my friend needs to find is, what is the radius of that fat head? I'm guessing it's probably around 7 feet, just judging by what it looks like when I wave my hand about 5' from side to side while only rising up 1. 5 inches. My friend is thinking he can tie a string that is the length of this radius between his router and something across his garage and create a nice smooth arc this way.

Any mathmeticians out there? In a nutshell: What is the radius of a circle (or if it's easier, the diameter divided by two) with a chord cutting across it that is 54" long and has a maximum thickness of 1. 5" from the middle of the chord line to the outer edge of the circle?

Thanks in advance, Andy

 

[guava]

r = (c^2+4h^2)/(8h),



r = radius

c = chord

h = height from the chord to the top of the arc.



assuming I understood the problem, the answer is 243. 75 inch radius.



I'm perfectly willing to be wrong on this and if I am, I hope someone will point it out.

 

[TPCDrafting]

Originally posted by guava

r = (c^2+4h^2)/(8h),



r = radius

c = chord

h = height from the chord to the top of the arc.



assuming I understood the problem, the answer is 243. 75 inch radius.



I'm perfectly willing to be wrong on this and if I am, I hope someone will point it out.

This is dead on. I just drew it in AutoCad. I drew 2 lines 54" long, offset 1 1/2" apart. Next drew a circle using 3 points. first point was the mid point of the upper line, the second two being the endpoints of the lower line (the chord).

The radius = 20'-3 3/4" (243. 75")

The circumference of the whole circle = 127'-7 1/2"

The arc length = 4'-6 1/8"



I have my units set to round to the nearest 1/16" of an inch.

 

[klenger]

AutoCAD is great for solving geometry problems. I never would have remembered the formula, or even knew where to find it.

 

[Rman]

I just looked it up in my book and checked it, right on the money by my numbers too.

 

[Andy Perreault]

Great, thank you, gentlemen! So 243. 75 inches divided by 12 comes out to 20. 3125 feet. I was way off with my guess of 7 feet!

Andy

 

[Rman]

I remember proving this formula in calculus II... . whata pain in the ***

 

[DaveN]

good job on the formula, but here's a way to cheat (kinda)



Just take a thin piece of wood (i. e. rip a 1x6 or whatever to 1/4" or so, you can do this with a circular saw), and have someone hold one end while you push the two ends together until you get to the curve you want, then trace it with a pencil.



It's not a true radius, but for cosmetic work it ends up looking fine.

 

[Rman]

Dave if that's the case just use some cheapo baseboards or something, they bend easy (most do at least). C'mon, we're all scientific here, we have to justify all that math crap they teach us in school for something.

 

[DaveN]

You know it's funny you said that rman -



I sometimes find myself overanalyzing a situation and using all sorts of complicated math etc (I'm an engineer by trade) just to try to justify paying all that money for an education that's rarely used... .



but I just wanted to post a simpler method for those who don't like math, sometimes when you're on the jobside you don't have acess to all the formulas and it's an easy way to cheat and get the job done.