Polygon Area Calculator

Frequently Asked Questions

How is a regular polygon's area found?

Area = (1/4)·n·s²·cot(π/n) for n sides of length s, or (1/2)·perimeter·apothem. The tool also reports perimeter and interior angle (n−2)·180/n.

What is the shoelace formula?

For any simple polygon given its vertices, area = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|. It works for irregular shapes without splitting them into triangles.

Does vertex order matter for the shoelace method?

Vertices must be listed in order around the polygon (clockwise or counter-clockwise). Out-of-order points give a wrong (often self-intersecting) area.

What is the apothem?

The perpendicular distance from a regular polygon's center to the midpoint of a side, r·cos(π/n). It is the radius of the inscribed circle.

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How This Calculator Works

For a regular polygon (all sides and angles equal) the area is ½ × perimeter × apothem, where the apothem is the perpendicular distance from the centre to a side. Splitting the polygon into n identical isosceles triangles gives apothem = s ÷ (2·tan(π÷n)) and circumradius = s ÷ (2·sin(π÷n)). The interior angle is (n − 2)×180° ÷ n and the exterior angle is 360° ÷ n. For an irregular polygon given by vertices, this tool uses the shoelace formula (Gauss's area formula): list the vertices in order, then Area = ½|Σ(x_iy_i+1 − x_i+1y_i)|. The cross-products accumulate signed trapezoid areas that telescope to the enclosed region; the absolute value removes orientation so clockwise and counter-clockwise both work. The sign before taking absolute value tells you the winding direction.

Worked Example: Regular Hexagon, side 10

With n = 6, s = 10: perimeter = 60. Apothem = 10 ÷ (2·tan(30°)) = 10 ÷ (2 × 0.57735) ≈ 8.6603. Area = ½ × 60 × 8.6603 ≈ 259.81 square units (exactly (3√3÷2)×10²). Interior angle = (6−2)×180°÷6 = 120°; exterior = 60°. For the irregular default pentagon with vertices (0,0), (4,0), (4,3), (2,5), (0,3): shoelace sum = (0·0−4·0) + (4·3−4·0) + (4·5−2·3) + (2·3−0·5) + (0·0−0·3) = 0 + 12 + 14 + 6 + 0 = 32, so Area = 32 ÷ 2 = 16 square units.

Where Polygon Area Is Used

Surveying & GIS: parcel and land-lot acreage is computed by shoelace from boundary coordinates.

CAD & computer graphics: mesh face areas, polygon fill, and collision regions.

Architecture & flooring: material estimates for non-rectangular rooms and tiling regular tessellations.

Robotics & gaming: navigation meshes and visibility polygons rely on signed area for orientation tests.

Common Mistakes & Edge Cases

Listing vertices out of order - the shoelace formula assumes consecutive boundary points; a scrambled order gives a self-intersecting (bowtie) shape with the wrong area.

Repeating the first vertex at the end - not needed; the formula closes the loop automatically with the modulo wrap.

Confusing apothem with circumradius for regular polygons; area uses the apothem (inradius), not the distance to a vertex.

Fewer than 3 vertices is not a polygon - rejected here.

Assuming the irregular formula needs convexity; the shoelace works for any simple (non-self-intersecting) polygon, convex or concave.

Related Concepts & Limits

As n → ∞ a regular polygon converges to a circle, and its area → πR² - Archimedes' method of inscribed polygons. The signed shoelace area underlies the orientation test in computational geometry (positive = counter-clockwise) and Pick's theorem links lattice-polygon area to interior and boundary points: A = i + b÷2 − 1. This tool handles simple polygons only; self-intersecting or multiply-connected regions need decomposition. Coordinates are parsed one vertex per line as "x, y"; up to roughly ten vertices keeps input manageable. Computations use double precision, accurate to about 15 significant digits - ample for surveying and CAD where input precision is the real limit.

Frequently Asked Questions

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