Polygon interior angles from ordered vertices

It works for convex and concave polygons, so a recessed corner can correctly produce a reflex angle greater than 180 degrees. The response preserves vertex order, reports all measurements in degrees, identifies the winding direction, and includes the theoretical total angle sum. That makes the result useful for geometry checks, drawing tools, coordinate-data validation, and reproducible automated workflows.

Enter an ordered polygon boundary

Provide at least three vertices as objects with finite numeric x and y coordinates. The order must follow the polygon boundary continuously, either clockwise or counterclockwise. Do not repeat the first point at the end because the calculator closes the last edge automatically. The coordinates describe a flat Cartesian plane; they may represent drawing units, meters in a projected coordinate system, pixels, or any other consistent linear scale. They are not interpreted as longitude and latitude on a curved Earth. A valid input must describe a simple polygon: neighboring vertices cannot be identical, no vertex may be duplicated elsewhere, the enclosed area must be nonzero, and nonadjacent edges cannot cross or touch. These rules matter because an interior angle is ambiguous when a ring folds over itself. If validation fails, the response identifies the relevant vertex or edge condition instead of returning plausible-looking measurements for invalid geometry. The maximum vertex count keeps the deterministic browser and API execution bounded.

Understand convex, concave, and reflex results

For each vertex, the calculator forms vectors from the current point toward its previous and next neighbors. Their dot product and cross product determine the local angle and turn direction. Signed polygon area establishes whether the complete boundary runs clockwise or counterclockwise. Combining the local turn with that winding is what distinguishes an ordinary convex corner from a concave corner whose interior is the reflex complement. Consequently, a convex rectangle returns four 90-degree values, while an inward notch can return an angle greater than 180 degrees at the recessed vertex. The angles_deg array uses exactly the same order as the input vertices, so index zero always describes the first supplied point. Results are rounded to ten decimal places to avoid distracting floating-point tails while retaining ample precision for common geometry work. Collinear boundary points may yield a 180-degree angle when they do not create an invalid overlap. The winding field describes orientation only; reversing the entire input order changes that field but preserves the corresponding geometric angle at each coordinate.

Check the total and use the output safely

Every simple polygon with n vertices has an interior-angle total of (n − 2) × 180 degrees, regardless of whether it is convex or concave. The total_sum_deg field reports that invariant directly, while vertex_count supplies n. This total is useful as a fast consistency check when importing geometry from CAD exports, digitized outlines, educational exercises, or custom drawing applications. The individual rounded values can differ from an independently accumulated floating-point sum by a tiny amount, so use total_sum_deg as the authoritative theoretical total rather than summing display values and demanding bit-for-bit equality. The capability is deterministic: the same ordered coordinates always produce the same JSON, and it uses no network service, random source, or current time. You can run it freely in the browser for interactive work or call the API for $0.002 per request when incorporating the calculation into a pipeline. Keep all coordinates in one planar reference system, and project geographic data before using this calculator when curvature would materially affect the intended geometry.