Polar Coordinate System

Pre-Calculus · Axiom Academy

LESSON The Polar Coordinate System Locate any point with a distance and an angle instead of a left-right, up-down grid. A polar point is written . To place it, do exactly two things from the center, called the pole : first rotate from the rightward reference ray by the angle , then walk outward r units in that direction. The watch below plots — swing out the angle, then march the distance. 2. The Frame: Pole and Polar Axis Every polar plot rests on two pieces. The animation builds them in order: the pole drops in first, then the polar axis shoots out to the right, and finally the rings and spokes fill in — the rings measure r , the spokes measure . The center of everything — the polar version of the origin (0, 0) . Every distance r is measured from here. The reference ray pointing right from the pole, like "due east" on a compass. Every angle is measured from it. Each circle is one distance from the pole: r = 1, 2, 3, 4 . They replace the up-down gridlines. Each ray is one heading out of the pole, drawn here every 30°. They replace the left-right gridlines. Here is the twist that surprises everyone: r is allowed to be negative . A negative radius does not mean "no distance" — it means walk |r| units in the opposite direction of the angle, straight back through the pole. The animation aims along , plots forward, then plots by reversing through the pole. Walk r units forward along the heading . The ordinary case.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.