Polygon Area Calculator

Enter the coordinates of each vertex of your polygon. Points will be connected in order, and the last point will connect back to the first.

Visual Representation

Grid: Each square = 1 unit

Area: Click "Calculate Area" to compute

How the Shoelace Formula Works

The Shoelace formula (also known as the surveyor's formula) calculates the area of a polygon by:

  1. Multiplying each x-coordinate by the next point's y-coordinate
  2. Subtracting the product of each y-coordinate with the next point's x-coordinate
  3. Taking half of the absolute value of the sum

Formula: Area = ½|Σ(xiyi+1 - xi+1yi)|

Related Geometry Calculators

  • Triangle Area Calculator
  • Circle Area Calculator
  • Rectangle Area Calculator
  • Perimeter Calculator
  • Distance Between Points Calculator
  • Midpoint Calculator
  • Coordinate Geometry Tools

Worked example: a triangle

Replace the default rectangle with three points: (0,0), (4,0), (0,3) — a right triangle with legs 4 and 3. The shoelace formula sums x_i·y_(i+1) − x_(i+1)·y_i across each edge, then halves the absolute total:

Sum = 0 + 12 + 0 = 12. Area = 12 / 2 = 6 square units. This matches the standard right-triangle formula (½ × base × height = ½ × 4 × 3 = 6), confirming the shoelace method works correctly on non-rectangular shapes, not just the rectangle it starts with.

Why the default rectangle gives 12

The four starting points — (0,0), (4,0), (4,3), (0,3) — form a 4×3 rectangle, so the plain width×height formula (4 × 3 = 12) already tells you the answer; running it through the shoelace formula is really a consistency check, since both methods must agree for a rectangle.

Point order matters

The formula assumes vertices are entered in order around the perimeter (clockwise or counterclockwise, either works) — entering them out of order, so the connecting lines cross, produces a self-intersecting shape and an area figure that no longer represents a simple polygon's true area.

Constraints

A minimum of 3 points is required. Every x and y field must contain a valid number, or the calculation is blocked with an explicit message rather than silently treating a blank field as zero.