Midpoint of polar coordinates calculator

It converts each polar point to Cartesian coordinates, averages the two horizontal coordinates and the two vertical coordinates, and converts that average back to polar form. You receive both representations, making the result easy to inspect or reuse. Choose degrees or radians once for the whole calculation, and the returned angle uses the same unit.

Enter two polar points in one angle unit

Describe the first point with radius_1 and angle_1, then describe the second with radius_2 and angle_2. Both radii must be zero or positive, while angles may be positive, negative, or larger than a complete turn. Select degrees when the angles are measured on a 360-degree circle, or radians when they are measured on a circle of 2π radians. The same selection applies to both inputs, which prevents an ambiguous mixture of units. A polar radius describes distance from the origin, so a negative radius is rejected rather than silently reinterpreted with a half-turn change in direction. Angles are otherwise accepted as written because trigonometric functions naturally treat coterminal values as the same direction. For example, 30 degrees, 390 degrees, and -330 degrees identify the same ray. If a point has radius zero, its supplied angle does not affect its location, but it must still be a finite number so the input contract stays predictable.

Understand why Cartesian averaging is necessary

A geometric midpoint is obtained by averaging positions, not by independently averaging polar radii and angles. Polar coordinates are nonlinear: angles wrap at the end of a turn, and changing a radius changes how much a direction contributes to the actual location. The calculator first applies x = r cos θ and y = r sin θ to each point. It then computes the midpoint coordinates as x = (x₁ + x₂) / 2 and y = (y₁ + y₂) / 2. Only after that linear average does it recover the midpoint radius with the Pythagorean distance from the origin and the midpoint angle with the two-argument arctangent. This distinction matters near the angle boundary. Two equally distant points at 350 degrees and 10 degrees have a midpoint on the positive horizontal axis, not at 180 degrees, even though a naive arithmetic average of their written angles suggests the opposite direction. Cartesian conversion resolves the wraparound automatically and also handles unequal radii correctly.

Read and reuse the normalized result

The response includes x and y for the Cartesian midpoint, plus radius, angle, and angle_unit for its polar form. The angle is normalized into a nonnegative complete turn: from zero up to, but not including, 360 in degree mode, or from zero up to, but not including, 2π in radian mode. This makes results easier to compare even when the input angles use negative or multi-turn notation. When the midpoint lies at the origin, its radius is zero and no geometric direction exists. The calculator returns angle zero by convention so the JSON remains finite, deterministic, and convenient for downstream systems. Numeric fields are rounded to twelve decimal places to suppress insignificant floating-point artifacts while preserving useful precision for ordinary geometry work. Because both coordinate forms are returned, you can use x and y directly in a drawing or vector calculation, or pass radius and angle into another polar-coordinate operation without repeating the conversion yourself.