# Circle Problem

#### Wendy

Joined Mar 24, 2008
21,906
I need to know the radius of a circle, that matches the following measurements, I think I have enough data to solve the problem but I am not sure how ro go about it. Any ideas? The dimension given are sloppy and approximations only.

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#### WBahn

Joined Mar 31, 2012
24,853
I need to know the radius of a circle, that matches the following measurements, I think I have enough data to solve the problem but I am not sure how ro go about it. Any ideas? The dimension given are sloppy and approximations only.

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b = (0.65")/2
a = (R - 0.20")
R² = a² + b²

R² = (R - 0.20")² + [(0.65")/2]²

R² = R² - 2(0.20")R + (0.20")² + (0.65")²/4

2(0.20")R = (0.20")² + (0.65")²/4

R = [(0.20")² + (0.65")²/4] / [2(0.20")]

R = 0.364"

Check:
a = 0.164"
b = 0.325"
R = 0.364"
Pythagorean sum checks good.

#### AlbertHall

Joined Jun 4, 2014
8,513

#### MrAl

Joined Jun 17, 2014
6,615
Hi,

Here is another way.

r^2=(x-h)^2+(y-k)^2

Now form three equations with dimensions extracted from the drawing shifting the x axis for convenience:
1. x=0, y=0.2
2. x=0.65/2, y=0
3. x=-0.65/2, y=0

That gives you three equations in the three unknowns r, h, and k.
Solving that set of three equations, we get two solutions:
r=-233/640,h=0,k=-21/128
r=233/640,h=0,k=-21/128

since h and k dont matter, we get the required radius:
r=233/640=0.3640625

If we stick with the actual dimensions instead of shifting the x axis, we get a different set of three equations using the actual x,y pairs:
0,0
0.65/2,0.2
0.65,0

and we get solutions:
r=-233/640,h=13/40,k=-21/128
r=233/640,h=13/40,k=-21/128

and so here we get the radius and the actual circle center offsets.

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#### WBahn

Joined Mar 31, 2012
24,853
I prefer my way
Since my way got me paid several hundred dollars by a high-end carpenter that wanted an equation that would let them calculate the needed radius of a circular arch out in the field, I kinda prefer mine. I didn't even expect to get anything for it -- I just spent worked out the solution and sent him a formula in where he could plug in the width of the opening and the height of the arch above the side walls and figured that was good. But he said that it saved him so much time and all of his fellow carpenters loved it so much, that he wanted to pay me a fraction of what the time that it saved them was worth so that I would be willing to do other things for them in the future.

#### ci139

Joined Jul 11, 2016
778
. -- the fuzzy construct is only for completing the schematic - the formulas are shown for the scaled projection of the original sketch /!\ = you should use the values of the not scaled H , W , M , R , etc. . . .
chk. R = ( W² / H + H ) / 2 = [ W = 0.65" / 2 , H = 0.2" ] = 233 / 640

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