Center — Definition, Formula & Examples

Center is the exact middle point of a shape. In a circle, the center is the single point that is the same distance from every point on the circle.

The center of a geometric figure is a point equidistant from all points on the boundary of the figure. For a circle with center $(h, k)$ and radius $r$ , every point on the circle lies exactly $r$ units from $(h, k)$ .

How It Works

To find the center of a circle, look for the point that is equally far from every spot on the edge. If you draw any straight line through the center of a circle from one side to the other, that line is called a diameter, and the center is its midpoint. Other shapes have centers too — a square's center is where its diagonals cross, and a sphere's center is the point equally far from every point on its surface.

Worked Example

Problem: A circle has a diameter with endpoints at (1, 2) and (5, 6). Find the center of the circle.

Step 1: The center is the midpoint of the diameter. Use the midpoint formula.

\text{Center} = \left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2}\right)

Step 2: Substitute the endpoint coordinates (1, 2) and (5, 6).

\text{Center} = \left(\frac{1 + 5}{2},\; \frac{2 + 6}{2}\right) = (3,\; 4)

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

Why It Matters

Knowing the center of a shape helps you measure distances, draw accurate diagrams, and solve problems about symmetry. In later courses like algebra and trigonometry, you will use the center of a circle to write its equation and graph it on a coordinate plane.

Common Mistakes

Mistake: Thinking the center of a circle is on the circle itself.

Correction: The center is inside the circle, not on its edge. Every point on the circle is a fixed distance (the radius) away from the center.