Hulknurk

Hulknurk is the Estonian term for a polygon. In geometry, a hulknurk is a plane figure formed by a finite sequence of straight line segments, called its sides, joined end to end to form a closed path. The endpoints of consecutive segments are the polygon's vertices. A hulknurk is simple if its sides do not cross, otherwise it may be self-intersecting.

An n-sided hulknurk has n sides and n vertices. The sum of interior angles equals (n − 2) × 180 degrees, and the exterior angles sum to 360 degrees. The perimeter is the sum of its side lengths. The area is shape-dependent and can be computed with the shoelace formula for arbitrary polygons.

A hulknurk is convex if every line segment joining two boundary points stays inside the figure; equivalently all interior angles are less than 180 degrees. If at least one interior angle is greater than 180 degrees, the hulknurk is concave. A regular hulknurk has equal side lengths and equal interior angles.

Diagonals connect non-adjacent vertices; a hulknurk with n vertices has n(n−3)/2 diagonals. Regular polygons have known properties: central angle 2π/n; area formulas include A = (n/4) s^2 cot(π/n) or A = (1/2) P a, where a is the apothem.

Polygons can be described by an ordered list of vertices (x_i, y_i) in either clockwise or counterclockwise order. The shoelace formula for the area is A = 1/2 |sum (x_i y_{i+1} − x_{i+1} y_i)| with (x_{n+1}, y_{n+1}) = (x_1, y_1).

Polygons appear in geometry, computer graphics, GIS, and CAD, where they serve as fundamental building blocks for planimetric shapes and tessellations.