Solve for Radius
Date: 09/05/97 at 15:02:19 From: Ron Culpepper Subject: Solve for radius Given three coplanar points, what is the formula for the radius?
From: Dr. Math [SMTP:firstname.lastname@example.org] Sent: Saturday, September 06, 1997 6:56 PM Subject: Re: Solve for radius Hi Ron, Do you mean the circle through the three points? If the points P1, P2, and P3 are in the (x,y)-plane, you could write the equation of the perpendicular bisector of the line joining P1 and P2 and do the same for the line joining P2 and P3. The intersection of the two perpendicular bisectors is the center of the circle. Then you could calculate the radius. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Date: 09/09/97 at 15:43:57 From: Ron Culpepper Subject: Re: Solve for radius Yes the circle through three points. Given p1,p2,p3 and B1 = (p1-p2)\2 B2 = (p1-p3)\2 B3 = (p2-p3)\2 what is the equation for the perpendicular bisector intersection or where can I find it?
Date: 09/15/97 at 12:28:40 From: Doctor Rob Subject: Re: Solve for radius The line connecting P1 and P2 is given by the two-point form of the equation of a line: (y-y1)*(x2-x1) = (y2-y1)*(x-x1) and similarly for P1 and P3, and for P2 and P3. The midpoint of P1P2 is ([x1+x2]/2,[y1+y2]/2). The perpendicular bisector must pass through this point and have slope the negative reciprocal of the slope of the line P1P2, that is, slope -(x2-x1)/(y2-y1). By changing the subscripts, the same holds true for the lines P1P3 and P2P3. This will allow you to write down the equations for the three perpendicular bisectors using the point-slope form of the equation of a line: if the point is (x0,y0) and the slope is m, then the equation is y = m*(x-x0)+y0. (You may have to deal with vertical lines, whose perpendicular bisectors are horizontal and so have slope zero, and horizontal lines, whose perpendicular bisectors are vertical and so have the form x = constant.) Now pick any two of these equations, and solve them simultaneously for x and y. That will give you the center of the circle. Find the distance of that point from any of the three given points to get the radius. Having said that, there is a formula for the radius you seek in terms of the lengths of the three sides. Let their lengths be a, b, and c. Let s = (a+b+c)/2, the semi-perimeter. Then the radius is R = a*b*c/(4*Sqrt[s*(s-a)*(s-b)*(s-c)]). You may recognize the square root in the denominator as the area of the triangle, according to Hero's Formula. You have to compute a, b, and c by using the distance formula. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.