A circle's equation, circumference, and area all come from the same two numbers: its center and radius. Drag the center and radius below, then sweep out a sector to see arc length and sector area scale with the angle.
The equation of a circle, (x − h)² + (y − k)² = r², comes directly from the distance formula: every point (x, y) on the circle sits exactly r units from the center (h, k). Reading off the center means watching the signs carefully — (x + 3) means h = −3, not 3.
Circumference (2πr) and area (πr²) describe the whole circle. A sector is just a fraction of that whole circle, cut out by a central angle — scale the circumference or area by (angle ÷ 360°) to get the arc length or sector area for any slice.
When a circle’s equation isn’t already in center-radius form — like x² + y² − 6x + 4y − 12 = 0 — completing the square on both the x-terms and y-terms separately converts it back into (x − h)² + (y − k)² = r², the same technique used to derive the quadratic formula.
Angles inside a circle follow their own rule. A central angle has its vertex at the circle’s center; an inscribed angle has its vertex somewhere on the circle itself. The Inscribed Angle Theorem says that whenever both angles intercept the same arc, the inscribed angle is always exactly half the central angle. The most useful special case follows directly from it: since a diameter always subtends a 180° central angle, any triangle inscribed in a circle with one side on a diameter must have a 90° angle opposite that diameter, every time.