How to Find the Center of a Circle — Definition, Formula & Examples

Finding the center of a circle is a geometric construction where you draw the perpendicular bisectors of two different chords — the point where those bisectors cross is the center.

The center of a circle is the unique point equidistant from every point on the circle. It can be located by constructing the perpendicular bisectors of any two non-parallel chords; their intersection is the center, since the perpendicular bisector of every chord passes through the center.

How It Works

Draw any chord across the circle, then construct its perpendicular bisector (a line at 90° through the chord's midpoint). Draw a second chord that is not parallel to the first, and construct its perpendicular bisector as well. The two bisectors will intersect at exactly one point — that point is the center. This works because a fundamental property of circles guarantees that the perpendicular bisector of any chord always passes through the center. You can use a compass and straightedge, or simply fold a circular cutout to create crease lines that act as perpendicular bisectors.

Worked Example

Problem: A circle passes through the points A(0, 0), B(6, 0), and C(6, 8). Find its center.

Step 1: Use chord AB from (0,0) to (6,0). Its midpoint is (3,0). Since AB is horizontal, its perpendicular bisector is the vertical line x = 3.

\text{Midpoint of } AB = \left(\frac{0+6}{2},\,\frac{0+0}{2}\right) = (3,\,0)

Step 2: Use chord BC from (6,0) to (6,8). Its midpoint is (6,4). Since BC is vertical, its perpendicular bisector is the horizontal line y = 4.

\text{Midpoint of } BC = \left(\frac{6+6}{2},\,\frac{0+8}{2}\right) = (6,\,4)

Step 3: Find where the two perpendicular bisectors intersect. The line x = 3 meets the line y = 4 at the point (3, 4).

\text{Center} = (3,\,4)

Answer: The center of the circle is at (3, 4).

Why It Matters

This construction appears in geometry courses whenever you need to circumscribe a circle around a triangle or find the circumcenter. It is also a practical skill — engineers and machinists use this method to locate the center of a circular part when no center mark exists.

Common Mistakes

Mistake: Drawing perpendicular bisectors of two parallel chords and expecting them to cross.

Correction: Parallel chords produce parallel perpendicular bisectors, which never intersect. Always choose two chords that point in different directions so their bisectors will meet at a single point.