What is the Shoelace formula and why is it called that?

The Shoelace formula (also called Gauss's area formula) computes polygon area from vertex coordinates by alternately multiplying and crossing x and y values in a pattern that visually resembles lacing a shoe. It gives the exact area for any simple polygon, regular or irregular, with any number of sides.

Does the Shoelace formula work for concave polygons?

Yes. The Shoelace formula correctly computes the area of any simple polygon, including concave (non-convex) polygons, as long as the sides do not cross each other. The absolute value in the formula handles both clockwise and counter-clockwise vertex orderings.

How many vertices can this calculator handle?

This calculator supports 3 to 8 vertices. A triangle requires exactly 3. Most practical irregular shapes (quadrilaterals, pentagons, hexagons, heptagons, octagons) are covered. For polygons with more than 8 sides, split the shape into sections and sum the areas.

What units does the area result use?

The area is expressed in square units, where one unit equals whatever unit you used for the coordinates. If you entered coordinates in metres, the area is in square metres. If you used feet, the area is in square feet. There is no unit conversion built in.

What is the difference between area and perimeter of a polygon?

Area is the total space enclosed inside the polygon boundary, measured in square units. Perimeter is the total length of all sides, measured in linear units. This calculator gives both: area via the Shoelace formula and perimeter by summing the Euclidean distances between consecutive vertices.

Can I use negative coordinates in this calculator?

Yes. Negative x or y coordinates are fully supported. Polygons in any quadrant of the Cartesian plane give the correct area and perimeter. The Shoelace formula handles negative coordinates naturally.

What happens if I enter vertices that cross each other (self-intersecting polygon)?

The Shoelace formula does not detect self-intersections. For a self-intersecting polygon, the formula returns the net signed area, which may not equal the total enclosed area you expect. Always enter vertices in a consistent order (fully clockwise or fully counter-clockwise) to get a correct result.

How do I find the area of a land plot or field using this calculator?

Survey the corners of the plot and record each corner as an (x, y) coordinate pair relative to a reference point. Enter the corners in order (walking around the boundary) into this calculator. The result is the area in the same units as your coordinates, typically square metres or square feet.

Irregular Polygon Area Calculator

Q: How do you calculate the area of an irregular polygon from coordinates?

Use the Shoelace formula: list all vertices in order, multiply each x by the next y and subtract each next x times the current y, sum all those products, take the absolute value, and divide by 2. This calculator does all of that automatically once you enter the vertex coordinates.

Q: How is the perimeter of an irregular polygon calculated?

The perimeter equals the sum of all side lengths. Each side length is the Euclidean distance between two consecutive vertices: square root of (x2 minus x1) squared plus (y2 minus y1) squared. This calculator adds up all those distances, including the closing side from the last vertex back to the first.

Enter up to 8 vertex coordinates to find the exact area and perimeter of any irregular polygon.

📐 What is an Irregular Polygon Area Calculator?

An irregular polygon area calculator computes the exact area enclosed by any polygon whose sides and angles are not all equal. Unlike a regular polygon (such as a square or equilateral triangle), an irregular polygon has vertices at arbitrary positions in the coordinate plane. The only requirement is that the polygon is simple, meaning its sides do not cross each other.

Real-world applications are extensive. Civil engineers use coordinate-based area calculations to determine the area of land parcels for construction permits and property valuations. Surveyors walk the boundary of a field, record each corner as a GPS coordinate, and then convert those readings into an area figure. Architects calculate floor plan areas from architectural drawings. Computer graphics programmers use polygon area formulas when rendering and clipping shapes on screen. In competitive programming and mathematics contests, the Shoelace formula appears regularly as an efficient O(n) algorithm for polygon area.

A common misconception is that irregular polygons require triangulation or complex numerical integration. In practice, the Shoelace formula gives an exact, closed-form result for any simple polygon in constant time per vertex. There is no approximation involved. Another misconception is that vertices must be in counter-clockwise order. The formula works equally well for clockwise ordering: the absolute value step removes the sign difference.

This calculator accepts up to 8 vertices, which covers the vast majority of practical irregular shapes encountered in education and professional work. It returns the area in square units and the perimeter in linear units, along with a detailed side-length table so you can verify each measurement individually.

📐 Formula

A = ½ × |∑_i=0^n-1 (x_i × y_i+1 − x_i+1 × y_i)|

A = area of the polygon in square units

n = total number of vertices

(x_i, y_i) = coordinates of vertex i (0-indexed)

Indices are cyclic: vertex n is the same as vertex 0 (closing the polygon)

Perimeter: P = ∑_i=0^n-1 √((x_i+1 − x_i)² + (y_i+1 − y_i)²)

Example: Triangle with vertices (0,0), (4,0), (0,3): A = ½|0×0−4×0 + 4×3−0×0 + 0×0−0×3| = ½|12| = 6 sq units

The formula is attributed to Carl Friedrich Gauss and is independently known as the Surveyor's formula in land measurement contexts. It is exact for any simple polygon (no self-intersections) regardless of convexity. For a self-intersecting polygon, the formula returns the net algebraic area, which may differ from the total enclosed area you intuitively expect.

📖 How to Use This Calculator

Steps

Enter vertex coordinates - Type the X and Y value of each corner into the numbered rows. Enter vertices in order, either all clockwise or all counter-clockwise around the polygon boundary.

Leave unused rows blank - If your polygon has fewer than 8 vertices, leave the extra rows empty. The calculator automatically skips any row where both fields are blank.

Click Calculate - Press the Calculate button to instantly see the area in square units and the perimeter in units.

Use the results - Read the area and perimeter from the results panel. The side-length table below shows the length of each individual side for further verification.

💡 Example Calculations

Example 1 - Right Triangle with Legs 5 and 12

Triangle with vertices (0, 0), (5, 0), (0, 12)

Apply Shoelace: sum = (0 x 0 - 5 x 0) + (5 x 12 - 0 x 0) + (0 x 0 - 0 x 12) = 0 + 60 + 0 = 60

Area = |60| / 2 = 30 sq units. Verify: (1/2) x base x height = (1/2) x 5 x 12 = 30. Correct.

Perimeter: leg1 = 5, leg2 = 12, hypotenuse = sqrt(25 + 144) = 13. Total = 5 + 12 + 13 = 30 units.

Area = 30 sq units, Perimeter = 30 units

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Example 2 - Irregular Pentagon (Real-World Land Plot)

Five-sided plot with vertices (0,0), (10,0), (12,6), (6,10), (−2,7)

Shoelace terms: (0x0 - 10x0) + (10x6 - 12x0) + (12x10 - 6x6) + (6x7 - (-2)x10) + ((-2)x0 - 0x7)

= 0 + 60 + (120 - 36) + (42 + 20) + 0 = 0 + 60 + 84 + 62 + 0 = 206

Area = |206| / 2 = 103 sq units. If coordinates are in metres, the plot covers 103 m².

Area = 103 sq units

Try this example →

Example 3 - L-Shaped Room (Concave Polygon)

Six-vertex L-shape: (0,0), (6,0), (6,3), (4,3), (4,5), (0,5)

Shoelace terms: (0x0-6x0) + (6x3-6x0) + (6x3-4x3) + (4x5-4x3) + (4x5-0x5) + (0x0-0x5)

= 0 + 18 + (18-12) + (20-12) + (20-0) + 0 = 0 + 18 + 6 + 8 + 20 + 0 = 52

Area = |52| / 2 = 26 sq units. Verify by splitting: 6x3 rectangle = 18, plus 4x2 rectangle = 8. Total = 26. Correct.

Area = 26 sq units

Try this example →

❓ Frequently Asked Questions

How do you find the area of an irregular polygon with coordinates?+

Use the Shoelace formula: list all vertices in clockwise or counter-clockwise order, multiply each x-coordinate by the next y-coordinate and subtract each next x times the current y, sum all those products, take the absolute value, and divide by 2. This calculator automates every step once you enter the vertex coordinates.

What is the Shoelace formula and how does it work?+

The Shoelace formula (Gauss's area formula) is A = (1/2)|sum of (xi times y(i+1) minus x(i+1) times yi)|, where i cycles through all vertices. The name comes from the crossing pattern of the multiplications, which visually resembles lacing a shoe. It provides an exact area for any simple polygon in linear time (one pass through the vertices).

Does this calculator work for concave polygons?+

Yes. The Shoelace formula correctly handles concave (non-convex) polygons as long as the polygon is simple (no sides cross each other). An L-shaped room, a star-shaped plot, or any irregular boundary without self-intersections will produce an accurate area result.

What order should I enter the vertices in?+

Enter vertices in a consistent order as you trace the polygon boundary, either all clockwise or all counter-clockwise. The absolute value in the formula ensures you get the same positive area regardless of which direction you choose. What matters is that you do not jump around (e.g., entering alternating vertices) since that would describe a self-intersecting shape.

Can I use negative x or y coordinates?+

Yes. Negative coordinates are fully supported. If your polygon straddles the origin or sits entirely in a negative quadrant, simply enter the negative values directly. The Shoelace formula handles all four quadrants without any adjustment.

How is the perimeter of an irregular polygon calculated?+

The perimeter is the sum of the Euclidean distances between every pair of consecutive vertices, including the closing segment from the last vertex back to the first. Each side length equals the square root of (x2 minus x1) squared plus (y2 minus y1) squared. This calculator shows every individual side length in the results table.

What units are used for the area and perimeter results?+

The results are in the same unit system as your coordinates. If you enter coordinates in metres, the area is in square metres and the perimeter is in metres. There is no built-in unit conversion. For land measurements, multiply by the appropriate factor (e.g., 1 m = 3.281 ft) before entering coordinates if you need a specific unit.

How do I calculate the area of a land plot from GPS coordinates?+

Convert your GPS coordinates (latitude/longitude) to a projected coordinate system such as UTM (Universal Transverse Mercator) to get x and y values in metres. Then enter those projected coordinates into this calculator. The Shoelace formula gives the area in square metres, which you can convert to acres (divide by 4047) or hectares (divide by 10,000).

What is the difference between a regular and an irregular polygon?+

A regular polygon has all sides equal in length and all interior angles equal (e.g., square, equilateral triangle, regular hexagon). An irregular polygon has sides and/or angles of different sizes. This calculator is designed for irregular polygons but also gives the correct result for regular ones entered as vertex coordinates.

Why does my area come out wrong when I enter the vertices out of order?+

Entering vertices out of boundary order (e.g., skipping from one side to the opposite side) creates a self-intersecting polygon, which the Shoelace formula treats as overlapping regions cancelling each other out. Always trace the polygon boundary continuously in one direction, vertex by vertex, to get the correct enclosed area.

Can I calculate the area of a polygon with more than 8 vertices?+

This calculator supports up to 8 vertices. For polygons with more sides, split the shape into smaller sub-polygons (each with 8 or fewer vertices), calculate each area separately using this tool, and add the results together. Make sure the sub-polygons cover the full area without overlap.

How accurate is the Shoelace formula?+

The Shoelace formula is mathematically exact for any simple polygon, not an approximation. The only source of error is floating-point arithmetic in the computer, which is typically accurate to 15 significant digits. For practical purposes the result is exact. No sampling, integration step size, or iteration count affects the answer.

🔗 Related Calculators

📌 Quick Tips

💡Enter vertices in order (all clockwise or all counter-clockwise). Mixing directions does not affect the area result but keeps the perimeter calculation correct.

💡Leave rows blank to skip them. The calculator ignores any row where both X and Y are empty.

💡Use decimal coordinates for real-world measurements. For example, enter 3.5 and 7.2 to represent metres or feet.

💡The Shoelace formula works for convex and concave polygons alike, as long as the polygon is simple (sides do not cross each other).

💡To find the area of a compound shape, split it into non-overlapping polygons, calculate each area separately, and add the results.