A circle is the set of all points in a plane at the same distance from a fixed center. That distance is the radius. The circle itself is the curve; the region inside is the disk. The distinction matters in geometry, though everyday speech often blurs it.

Circles have fascinated mathematicians for millennia. The ratio of circumference to diameter is pi, an irrational number approximately 3.14159. The area is pi times the radius squared. These two formulas appear in nearly every geometry course and in countless engineering calculations.

Circles appear in wheels, gears, orbits, ripples, and clock faces. They have perfect symmetry: every diameter is a line of symmetry, and every rotation about the center maps the circle onto itself. In analytic geometry, the equation (x − h)² + (y − k)² = r² describes a circle with center (h, k) and radius r. Simple to define, endlessly useful.

• Set of points equidistant from a center
• Radius is the distance from center to edge
• Circumference = 2πr; area = πr²
• Perfect rotational symmetry