Polygon Area Calculator

The polygon area calculator finds the area of any polygon from coordinates. Enter points, see the plot, and read the math.

Overview

Enter the coordinates of each vertex in order. The tool connects points in that exact sequence. It closes the shape by linking the last point to the first point. Click Calculate Area to get the result in square units. A clear diagram appears on a grid, and each square equals one unit. The shoelace formula steps display for full transparency. A recent update corrected a calculation error and improved clarity.

How It Works

This polygon area calculator uses the shoelace formula, also called the surveyor formula. It computes the area from your ordered coordinates.

1. Multiply each x coordinate by the next point’s y coordinate.
2. Multiply each y coordinate by the next point’s x coordinate.
3. Sum both sets of products and subtract the second from the first.
4. Take the absolute value and divide the result by two.

Formula: Area = 1/2 × |Σ(x_iy_i+1 − x_i+1y_i)|

Notes and limits

• Enter points in order around the shape for a correct area.
• Three points or more form a valid polygon.
• The last point links back to the first point by design.
• The displayed area is a positive number in square units.
• Self crossing shapes return a net area, which can differ from a union area.
• Use one unit system for all points for consistent results.
• Very large values can fall outside the visible grid window.
• Colinear points reduce area and can lead to zero area shapes.

Step by step instructions

Step 1: Enter Coordinates

Type an x coordinate and a y coordinate for each point. Use decimals as needed.

• x coordinate field: Enter the horizontal value of the point.
• y coordinate field: Enter the vertical value of the point.
• Order: Enter points in the path you would draw the edges.
• Units: Pick one unit and use it for all points.

Inputs explained

• Add Point: Add another vertex to your polygon.
• Remove Point: Remove the last vertex while three or more remain.
• Clear All: Clear every field and reset the diagram and steps.
• Calculate Area: Compute and display the area and the math steps.

Quick example

Points (0, 0), (4, 0), (4, 3), and (0, 3) form a rectangle. The polygon area calculator returns 12.0000 square units for this case.

Step 2: Adjust the Points

Need another vertex, click Add Point. Want fewer vertices, click Remove Point. Keep at least three points for a valid polygon. Edit any number as needed, and watch the diagram change as you type.

Step 3: Calculate the Area

Click Calculate Area to run the shoelace formula on your points. The result shows as a positive value with four decimal places. The steps panel lists each product, the sums, and the final half value. The diagram shades the polygon for a fast visual check.

Why Use Our Polygon Area Calculator?

This polygon area calculator gives clear math and a precise plot. You see the steps, the area, and the shape in one place. The method is standard and taught in many math and engineering courses. The result rounds to four decimals for clean reporting.

What the results mean

• Area: The polygon’s area in square units, shown to four decimals.
• Shoelace steps: The exact products and sums used in the formula.
• Diagram: A plotted shape on a grid where each square equals one unit.

Common use cases

• Class assignments: Check polygon area work with clear steps.
• Land planning: Estimate a lot outline from corner coordinates.
• Floor plans: Measure rooms or zones from corner points.
• Computer graphics: Validate polygon area in a drawing or mesh.
• Geographic information systems: Compute area after converting to flat coordinates.
• Research: Compare area across shapes built from measured points.

Tips for best results

• Enter points in a single loop without jumps across the shape.
• Use consistent units, like meters or feet, across all coordinates.
• Check the diagram for the intended shape before you compute.
• Concave polygons work as well as convex polygons with the same method.

This tool keeps geometry work simple and reliable. Add points, view the plot, and read the math. You get a clean area figure you can trust.