Circle

A circle is the set of all points in a plane that are a fixed distance — the radius $r$ — from a fixed point called the center.

Standard equation in the coordinate plane (center $(h, k)$, radius $r$):
$(x - h)^2 + (y - k)^2 = r^2$

Key measurements:

• Diameter: $d = 2r$ (longest chord, passes through center)
• Circumference: $C = 2\pi r = \pi d$
• Area: $A = \pi r^2$

The constant $\pi \approx 3.14159$ is the ratio of any circle's circumference to its diameter — it's the same for every circle, which is why $\pi$ shows up everywhere in geometry, trigonometry, and physics.

Important parts: a chord is any segment with endpoints on the circle; a tangent touches the circle at exactly one point and is always perpendicular to the radius at that point.