How to Find the Equation of a Circle, Given Only Two Endpoints of Its Diameter

You might think you can't do much with two points plotted on the xy-plane. However, the opposite is true. If you are given that these two points are endpoints of a circle's diameter, you can find the equation of the circle. This article examines the procedure one can use to do exactly that, given two points: (x1, y1) and (x2,y2).

Things You'll Need

• pen/pencil
• paper
• calculator
• general equation of a circle
• Midpoint Formula
• Distance Formula

Instructions

1. Write the general equation of a circle. It is (x-h)^2+(y-k)^2=r^2, where (h,k) is the center of the circle and r is the circle's radius.

2. Use the Midpoint Formula to find the circle's center. The Midpoint Formula is ((x1+x2)/2, (y1+y2)/2) or (average of the x-coordinates, average of the y-coordinates). The average of the x-coordinates is h, and the average of the y-coordinates is k.

3. Use the Distance Formula to find the distance between the endpoints on the circle's diameter. I have an eHow article that explains how to apply the formula. The equation is: d=sqrt[(x2-x1)^2+(y2-y1)^2]. (x1,y1) and (x2,y2) are the coordinates of the two points. The value you find is the diameter, d. You want the radius, r, however, so just divide the diameter by 2.

4. Square the result for r you obtained in Step 3. This is r^2, and now you've completed the equation.

Tips & Warnings

• Steps 2 and 3 can be switched if you want.

• Be careful with the plus and minus signs.

Resources

• Read this Article in Spanish