Area of Irregular Polygons

Introduction

I just thought I would share with you a clever technique I once used to find the area of general polygons.

The polygon could be regular (all angles are equal and all sides are equal) or irregular

Regular	Irregular

The Example Polygon

Let's use this polygon as an example:

Coordinates

The first step is to turn each vertex (corner) into a coordinate, like on a graph:

Area Under One Line Segment

Now, for each line segment, work out the area down to the x-axis.

So, how do we calculate each area?

Average the two heights, then multiply by the width

Example: For the shape highlighted above, we take the two heights (the "y" coordinates 2.28 and 4.71) and work out the average height:

(2.28+4.71)/2 = 3.495

Work out the width (the difference between the "x" coordinates 2.66 and 0.72)

2.66-0.72 = 1.94

The area is width×height:

1.94 × 3.495 = 6.7803

Add Them All Up

Now add them all up!

But the trick is to add when they go forwards (positive width), and subtract when they go backwards (negative width).

If you always go clockwise around the polygon, and always subtract the first "x" coordinate from the second, it works out naturally, like this:

You can also go the other direction. If you get a negative area just make it positive.

And it looks like this:

So that's it! The area is 8.3593

Area of Polygon Tool

Glad you read this far! You are rewarded with a link to the Area of a Polygon Drawing Tool that can do all of this for you. It also accepts manual entry of coordinates.

Formulas

Let's play with some formulas!

Let's start with a simple triangle, with coordinates (x₁, y₁), (x₂, y₂), (x₃, y₃)

Multiplying widths by average heights gets:

Area = (x₂−x₁)(y₁+y₂)/2 + (x₃−x₂)(y₂+y₃)/2 + (x₁−x₃)(y₃+y₁)/2

But we can rewrite it! Let's expand the whole thing so we can do some surgery on it:

2×Area = x₂y₁ + x₂y₂ − x₁y₁ − x₁y₂ + x₃y₂ + x₃y₃ − x₂y₂ − x₂y₃ + x₁y₃ + x₁y₁ − x₃y₃ − x₃y₁

Collect like terms:

2×Area = x₂y₁ + x₂y₂−x₂y₂ + x₁y₁−x₁y₁ − x₁y₂ + x₃y₂ + x₃y₃−x₃y₃ − x₂y₃ + x₁y₃ − x₃y₁

Simplify:

2×Area = x₂y₁ − x₁y₂ + x₃y₂ − x₂y₃ + x₁y₃ − x₃y₁

Rearranging the terms (and changing signs) we get the so-called "shoelace formula":

−2×Area = x₁y₂ + x₂y₃ + x₃y₁ − x₂y₁ − x₃y₂ − x₁y₃

Why is it called the Shoelace Formula?

Write the coordinates in a column, repeating the first vertex at the bottom:

Multiply along the downward diagonals (x₁y₂, x₂y₃, x₃y₁) and add them.

Multiply along the upward diagonals (x₂y₁, x₃y₂, x₁y₃) and subtract them.

The criss-cross pattern looks just like tied shoelaces!

In General

In general (for an n-sided simple polygon), using absolute values || we have:

Area = ½| (x₂−x₁)(y₁+y₂) + (x₃−x₂)(y₂+y₃) + ... + (x₁−x_n)(y_n+y₁) |

or the Shoelace version:

Area = ½| x₁y₂ + x₂y₃ + ... + x_ny₁ − x₂y₁ − x₃y₂ − ... − x₁y_n |

Using JavaScript the essential line (that gets summed up over all points) is:

let a1 = (pt1.x - pt0.x) * (pt0.y + pt1.y);  // for trapezoid

let a1 = (pt0.x * pt1.y - pt0.y * pt1.x);  // for shoelace

I calculated the area of a regular pentagon 1,000,000 times and both methods took about 43ms. I imagine the browser came up with the same underlying steps for both cases!