Finding the radius of a circle from three points on circle

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fro_ingot

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Is there a formula or process I can use to determine a radius when the only information supplied is three points on the circumference in the same hemisphere [A,B,C], the distance between the farthest points [AC], and the distance perpendicular from the center of line AC [D] to point ?

Example: The distance from A to C equals 65", and the distance from D to B equals 15".
 
Re: Finding the radius of a circle

If I am interpreting correctly.

AC is a long chord with length 65" and BD is a middle ordinate with length 15".

The radius is R=OA=OC. That's what we need. I will use @ for the angle subtended by the chord.

The formula for a long chord is \(\displaystyle 2Rsin(@/2)=65\)

The formula for the middle ordinate is \(\displaystyle R(1-cos(@/2))=15\)

You have two equations with two unknowns. Can you solve for R and @?.
 

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Thanks for your quick response.
Well, you lost me with the terms "chord" and "ordinate" (you know, old guy vs. new math), but the drawing you supplied is exactly what we're trying to determine. If I understand your query, I guess that 2 unknowns are one too many...
I could say that DO=R-15, but that doesn't help, does it :?

(I'm tickled that I even remember that sin & cos mean sine and cosine, but I can't remember what they do. Maybe if I still had my TI30 from the day...)
 
fro_ingot said:
Thanks for your quick response.
Well, you lost me with the terms "chord" and "ordinate" (you know, old guy vs. new math), but the drawing you supplied is exactly what we're trying to determine. If I understand your query, I guess that 2 unknowns are one too many...
I could say that DO=R-15, but that doesn't help, does it :?

(I'm tickled that I even remember that sin & cos mean sine and cosine, but I can't remember what they do. Maybe if I still had my TI30 from the day...)
Click to expand...
You have two equations and two unknowns - exactly what you need. However, the solution is little complicated.

Before I get into that - may I ask if this is a "book" problem or "practical" problem?

If you are allowed to use compass and ruler - you can solve this quickly.
 
Write the equations of the perpendicular bisectors of the line segments AB & BC.
The point where those two lines intersect is the center of the circle.
The distance from the center to any one of the points is the radius.
 
fro_ingot said:
Practical - we can use whatever tools are available
Click to expand...

I assume you know where to locate A and C and you know how to draw a perpendicular bisector of a line segment with compass and ruler..

(1) First we need to locate the point B (in case it is given then go to step 4)

(2) Bisect perpendicularly AC.

(3) Locate your point B on that perpendicular bisector - at a height 15 units

(4) Join AB and BC

(5) Draw perpendicular bisectors of AB and BC and extend those till those intersect.

(6) If there is any justice in the world - all your perpendicular bisectors (step 2 and 5) will meet at a point. That point is the center of your circle
 
Is there a formula or process I can use to determine a radius when the only information supplied is three points on the circumference in the same hemisphere [A,B,C], the distance between the farthest points [AC], and the distance perpendicular from the center of line AC [D] to point ?

Example: The distance from A to C equals 65", and the distance from D to B equals 15"

If C = the chord length AC and h = the segment height BD, then the radius R = [c^2 + 4h^2]/8h.


R = sector radius
c = chord length
d = distance from center to chord
h = height of segment
s = arc length
µ = sector entral angle, rad.
Ast = segment area
Asr = sector area

Given R and h: µ = 2arccos[(R-h)/R]

Given R and s: µ = s/R

Given R and d: µ = 2arccos[d/R]

Given R and c: µ = 2arsin[c/2R]

Given d and h: R = d + h

Given s and c: c/2s = [sin(µ/2)/µ]

Given s and d: d/s = [cos(µ/2)/µ]

Given c and h: R = [c^2 + 4h^2]/8h

Given c and d: R = sqrt[(4d^2 + c^2)/2]

Given h and s: h/s = [1 - cos(µ/2)]/µ

Given h and µ: R = h/cos(µ/2)

Given µ and d: R = d/cos(µ/2)

Given c and µ: R = c/2sin(µ/2)

s = Rµ

c = 2Rsin(µ/2)

d = Rcos(µ/2)

h = R[1 - cos(µ/2)]

Ast = R^2[µ - sin(µ)]/2

Asr = µR^2/2
 
Hello, fro_ingot!

Is there a formula or process I can use to determine a radius when the only information supplied
is three points on the circumference in the same hemisphere [A,B,C], the distance between the
farthest points [AC], and the distance perpendicular from the center of line AC [D] to point ?

Example: The distance from A to C equals 65", and the distance from D to B equals 15".
Click to expand...

\(\displaystyle \text{It seems that }\Delta ABC\text{ is isosceles, with base 65 and height 15.}\)
\(\displaystyle \text{From the dimensions, }\Delta ABC\text{ is inscribed in a semicircle.}\)

Code: 
                B
              * * *
          *     |15   *
      A * - - - * - - - * C
       *       D|     *  *
                |y  * R
      *         | *       *
      *         *         *
      *         O         *

       *                 *
        *               *

\(\displaystyle \text{We have: }\:BD = 15,\;AC = 65\quad\Rightarrow\quad DC = \frac{65}{2}\)

\(\displaystyle \text{Let: }\:DO = y,\;OB = OC = R\)


\(\displaystyle \begin{array}{cccc}\text{From radius }OB\text{, we have:} & y + 15 & = & R \\\text{From right triangle }ODC\text{, we have:} & y^2 + \left(\frac{65}{2}\right)^2 & = & R^2 \end{array}\)

\(\displaystyle \text{Solve the system of equations.}\)

 
Somehow, I misread the problem to be "locating" the center as opposed to"finding the measure of radius".

If only the measure of radius is needed - Soroban's method is best.

Square the first equation to get:

\(\displaystyle y^2 + 30y + 225 = R^2\)

Subtract equation (2)

\(\displaystyle 30y + 225 - 1056.25 = 0\)

\(\displaystyle y = 27.70833333\)

\(\displaystyle R = 15 + 27.70833333 = 42\frac{17}{24} = \~ 42\frac{45}{64}\)
 
The perpendicular bisector of a chord contains the center of the circle. If you extend segment BD through the center to meet the circle at another point E on the circle, BE will be a diameter of the circle.

A nice fact from geometry will make this problem fairly easy to solve:

If two chords intersect in a circle, the product of the segments of one of them is equal to the product of the segments of the other. We have chords AC and BE intersecting at D. So,
(AD)*(DC) = (BD)*(DE)

AC = 65, and D is the midpoint of AC. This means that AD and DC each have length 65/2, or 32.5.

We know that BD = 15.
Let r = radius of the circle. BE = 2r. BE - BD = DE. DE = 2r - 15.

Ok...substitute the values we know into

(AD)*(DC) = (BD)*(DE)
32.5*32.5 = 15(2r - 15)

That should be easy enough to solve for r.
 
Awesome. Thanks for all the help, guys. Now I need to study it so we can not only understand it, but remember it!
Nice to know there's a site where so many people are willing to help in such a short period of time! I'll reccommend the site!
 
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