How to Find Circumcenter Angles

All triangles have circumcenters.

A triangle's circumcenter exists where the perpendicular bisectors of each of its sides intersect. A perpendicular bisector is a line that passes through the midpoint of a line segment at a 90-degree angle. Finding the angles formed by the lines intersecting at a triangle's circumcenter is a simple matter of drawing the lines, then measuring the resulting angles with a protractor.

Things You'll Need

• Protractor
• Ruler

Instructions

1. Measure each side of a triangle. Put a mark precisely at the midpoint of each side.

2. Draw a perpendicular bisector through one side. Align the flat edge of your protractor with one of the triangle's sides, with the center of the edge directly over the mark at the midpoint of that side. Make a mark on your paper at the 90-degree point. Draw the perpendicular bisector to that side by connecting the 90-degree mark to the midpoint of the triangle's side and extending the line through the middle of the triangle.

3. Repeat the previous step for each side. Extend each line as necessary so that they intersect with one another. The point where they intersect is the circumcenter of your triangle. You will have created 6 angles surrounding the circumcenter.

4. Number the angles 1 through 6.

5. Measure the first angle. Place the center of the protractor directly over the triangle's circumcenter. Line up the protractor's straight edge with the line forming the "bottom" of angle 1. Follow the line forming the "top" of angle 1 to the mark where it meets the curved edge of the protractor. The reading at that mark is the measurement of the angle. If the mark reads 25 degrees, then it is a 25-degree angle.

6. Repeat the previous step for the remaining 5 angles.

References

• John Page/Math Open Reference: Circumcenter of a Triangle