Irregular Polygon Area Calculator - Shoelace, Perimeter, Centroid

An irregular polygon area calculator is a coordinate-geometry tool that finds the area, perimeter, centroid, and vertex count of any simple polygon from a typed list of x,y vertex coordinates. It applies the shoelace formula (also called Gauss's area formula) so you do not have to break the shape into triangles by hand. Type the vertices in order around the boundary and the calculator handles the signed-area math, the edge-length total, and the centroid.

• • Surveying and land parcels: Compute the area of an irregular land parcel from a list of boundary coordinates recorded by GPS or a total station.
• • Concave floor plans: Find the floor area of an L-shaped or T-shaped room where the regular polygon formulas no longer apply.
• • CAD and GIS verification: Cross-check a closed polyline's area in a CAD or GIS file before exporting measurements to a report.
• • Homework and competition geometry: Solve contest problems that give only vertex coordinates and expect a fast, exact shoelace answer.

The shoelace formula works for any simple polygon, which means a closed shape whose edges do not cross each other. The vertices can be convex or concave and the side lengths can be different, which is what makes the formula more useful for real drawings than the regular-polygon area equation. The square units of the result match the square units of the input coordinates, so the same formula works in centimeters, feet, surveyor's rods, or any other length unit.

Use this irregular polygon area calculator whenever a shape is defined as a list of coordinate pairs. If the shape has only three vertices, the result still applies, but a dedicated triangle calculator is a faster shortcut for the area and angles of a single triangle.

When every side and angle happens to be equal, the regular Polygon Area Calculator returns the same area from a single side length without typing each coordinate.

The calculator reads the typed vertex list, parses each line into an (x, y) pair, and then runs the shoelace formula in one pass around the polygon. The same loop tallies the edge lengths for the perimeter and accumulates the centroid sums.

• x_i, y_i: Coordinates of the i-th vertex, listed in order around the polygon (clockwise or counterclockwise both work).
• x{i+1}, y{i+1}: Coordinates of the next vertex, with the last vertex treated as adjacent to the first.
• A: Reported area, taken as the absolute value of the signed sum so the result is always positive.
• C_x, C_y: Centroid coordinates, computed from the same cross-product terms (1/(6A)) * sum((x_i + x{i+1}) * cross).
• P: Perimeter, the sum of the Euclidean distances between consecutive vertices including the closing edge.

The shoelace sum is also called Gauss's area formula or the surveyor's formula, and it works for convex and concave polygons alike. If the signed sum comes out negative, the vertices were listed clockwise; if it comes out positive, they were counterclockwise. The calculator takes the absolute value either way so the headline area is always positive.

The centroid uses the same cross-product terms. C_x = (1 / (6 * A_signed)) * sum((x_i + x{i+1}) * (x_i * y{i+1} - x{i+1} * y_i)), and C_y follows the same pattern with y. When the signed area is effectively zero (collinear points), the calculator falls back to the simple mean of the vertices so the centroid stays defined.

According to Wikipedia, the area of a simple polygon with vertices (x_1, y_1), ..., (x_n, y_n) is 0.5 times the absolute value of the shoelace sum over consecutive vertex pairs.

When the polygon has only three vertices, Triangle Calculator returns the same area using Heron's formula or two edge lengths and the included angle, which is a useful sanity check for the smallest shoelace case.

Four ideas hold the shoelace formula together. Knowing them makes the result easier to interpret when the polygon gets complicated.

The shoelace formula handles concave (non-convex) shapes by treating the inward dents correctly. The cross-product sum adds a positive contribution for a counterclockwise edge and a negative contribution for one that bends back into the polygon, so the result still equals the true enclosed area.

When the polygon turns out to be a rectangle, Length Width Area Rectangle Calculator gives the same area in one multiplication once you know the length and width.

Type the polygon vertices as (x, y) pairs in order, and the calculator returns area, perimeter, centroid, and vertex count.

1. 1 List the vertices in order: Walk around the polygon boundary once and record each (x, y) pair. Direction (clockwise or counterclockwise) does not matter for the reported area.
2. 2 Type the coordinates: Enter one (x, y) pair per line, separated by a comma. Semicolons or extra spaces are also accepted, so spreadsheet exports paste in cleanly.
3. 3 Read the results: Watch the area, perimeter, centroid, and vertex count update as you type. The vertex count tells you whether all the lines were parsed.
4. 4 Use the value in your project: Copy the area into a worksheet, CAD dimension, or assignment answer. The square units match the units used for the input coordinates.

For a 5-vertex lot drawn as (0,0), (12,0), (12,5), (8,9), (0,5), the calculator returns area = 84 square units, perimeter = 36.60, centroid = (6.19, 3.60), and vertex count = 5. That is enough to confirm the surveyor's plan and to set up a tax assessment.

When you also need the area of a different shape in the same project, Area Calculator covers common circles, rectangles, and triangles in one place.

The shoelace formula runs in one pass, which keeps the calculator fast even for polygons with many vertices. The same loop also feeds the perimeter and the centroid, so you get a full description of the shape from one input.

• • Handles any simple polygon: Convex, concave, triangle, or 20-vertex plot, the same formula and the same calculator cover them all without switching tools.
• • Works in any length unit: The area is in the same square units as the input coordinates, so the calculator is unit-agnostic. Meters, feet, and surveyor's rods all work the same way.
• • Returns a full shape description: Area, perimeter, centroid, and vertex count are all in one result panel, so you do not have to chain multiple calculations together.
• • Tolerant of messy input: Newlines, semicolons, extra spaces, and parentheses are all accepted, which makes copy-pasting from a spreadsheet or a CSV file painless.
• • Concave-friendly: The shoelace sum handles inward dents and L-shapes without you having to split the shape into triangles yourself.

For surveyors and GIS analysts the main win is that the formula runs on whatever vertex list is already in the file. For students and competition participants the win is that the answer comes out in one step, with the centroid included for free.

If the polygon happens to be a regular hexagon, switching to Hexagon Calculator skips the coordinate entry because the side length alone gives the area, perimeter, and apothem in one pass.

The shoelace formula is exact, but the result still depends on the quality of the input. These are the most common sources of a wrong answer.

• • Self-intersecting polygons (bow-tie shapes) are not supported. The shoelace sum still returns a number, but the absolute value no longer matches the actual enclosed area.
• • The calculator uses planar (Euclidean) geometry. Coordinates that represent points on a sphere or another curved surface need a different area formula.

If your polygon might cross itself, split it into non-overlapping parts first and add the individual shoelace results. The same advice applies if some of the vertices sit on top of one another, because the cross-product terms cancel.

According to Wolfram MathWorld, the shoelace formula gives the area of any simple polygon in the plane from its ordered list of vertex coordinates.

For a round shape that would otherwise need a long list of vertex coordinates, Circle Calculator returns the same area from a single radius input without the coordinate step.