Polygon Area Calculator

What This Polygon Area Calculator Does

This calculator computes the area of any simple polygon (a flat shape made from straight line segments that do not cross) from its vertex coordinates. You can enter coordinates for triangles, rectangles, convex polygons, and concave polygons, as long as the boundary does not self-intersect and the points are listed in order around the shape.

The tool uses the shoelace formula (also called Gauss's area formula). Internally it works with a signed area that depends on the orientation (clockwise or counterclockwise) of your vertices, then reports the final area as a positive value in squared units that match your input coordinates.

Typical uses include:

• Estimating the area of irregular land parcels from survey points
• Measuring areas in CAD drawings or GIS layers after exporting coordinates
• Checking the area of arbitrary polygons in simulations or computer graphics
• Classroom work on coordinate geometry and polygon area formulas

Shoelace Formula for Polygon Area

Suppose your polygon has n vertices listed in order around the boundary:

(x₁, y₁), (x₂, y₂), …, (x_n, y_n)

The shoelace formula for the signed area A is:

Here, the indices wrap around so that vertex n + 1 is the same as vertex 1. The vertical bars mean you take the absolute value: the formula naturally produces a positive area for one orientation (usually counterclockwise) and a negative area for the opposite orientation, but the enclosed area is always positive.

The name “shoelace” comes from the crisscross multiplication pattern you would see if you wrote the coordinates in two columns and drew lines between them, similar to lacing a shoe.

How to Use the Polygon Area Calculator

1. Gather your vertices.

   List all corner points of the polygon in order around the boundary. You can go clockwise or counterclockwise, but you must not jump back and forth across the shape.

2. Choose a coordinate system and units.

   Your coordinates can be in meters, feet, centimeters, or any other consistent unit. The area result will then be in the corresponding squared unit (square meters, square feet, etc.).

3. Enter coordinates one per line.

   In the input box, type each vertex on its own line as x,y or x y. The calculator accepts commas or spaces as separators.

   For example:

   0,0
   4,0
   4,3
   0,3

4. Compute the area.

   Click the button to compute. The tool will parse your coordinates, apply the shoelace formula, and display the polygon area. If the vertices are valid, orientation does not matter because the calculator reports the absolute area.

5. Interpret the result.

   Match the units of the result to your input. If your coordinates were in meters, the result is in square meters; if in feet, the result is in square feet.

Worked Example: Rectangle Using Coordinates

Consider a rectangle that is 4 units wide and 3 units tall. Place its vertices at:

• (0, 0)
• (4, 0)
• (4, 3)
• (0, 3)

We will list them in order and apply the shoelace formula.

1. Write down the coordinates in order and repeat the first point at the end:

   (x1, y1) = (0, 0)
   (x2, y2) = (4, 0)
   (x3, y3) = (4, 3)
   (x4, y4) = (0, 3)
   (x5, y5) = (0, 0)

2. Compute the sum of x_i × y_i+1:
   • 0 × 0 = 0
   • 4 × 3 = 12
   • 4 × 3 = 12
   • 0 × 0 = 0

   Total: 0 + 12 + 12 + 0 = 24

3. Compute the sum of y_i × x_i+1:
   • 0 × 4 = 0
   • 0 × 4 = 0
   • 3 × 0 = 0
   • 3 × 0 = 0

   Total: 0 + 0 + 0 + 0 = 0

4. Apply the shoelace formula:

   A = (1/2) × |24 − 0| = (1/2) × 24 = 12

The rectangle’s area is 12 square units, which matches the usual formula: width × height = 4 × 3 = 12.

If you enter the same coordinates into the calculator (one vertex per line, in order), it will display an area of 12. If you reverse the order of the vertices (making them clockwise instead of counterclockwise), the internal signed area would change sign, but the reported area remains 12 because the tool uses the absolute value.

Interpreting the Calculator’s Results

When you run the calculation, the tool returns a single number representing the polygon’s area. Keep these points in mind when interpreting the output:

• Units: The calculator does not impose units. Your coordinates decide the units. If you input meters, you get square meters; if you input feet, you get square feet.
• Positive area: Even though the shoelace formula produces a signed area (positive or negative depending on orientation), the calculator always reports the positive area. You do not need to worry about clockwise versus counterclockwise input.
• Scale dependence: Scaling all coordinates by a factor of k scales the area by k². Doubling all coordinates multiplies area by 4, and so on.
• Concave polygons: Concave shapes (that bend inward) are supported as long as the vertices follow the boundary in order and the polygon does not cross itself.

Comparison: Shoelace Method vs. Other Area Methods

The shoelace formula is one of several ways to compute polygon area. The table below compares this calculator’s method with a few common alternatives.

Use this polygon area calculator when you already have vertices or can easily record them. For very simple shapes like rectangles or basic triangles, a dedicated calculator or direct formula may be quicker, but for complex outlines with many corners, the shoelace approach is usually more efficient and less error-prone.

Assumptions and Limitations

For accurate results, this calculator relies on several important assumptions:

• Simple polygons only: The shape must be simple — its edges do not cross or self-intersect. Self-intersecting “bows” or “figure-eight” shapes are not supported.
• Vertices in order: Enter vertices in sequence around the boundary (clockwise or counterclockwise). Jumping around the shape or mixing the order will generally produce incorrect areas.
• Minimum of three points: At least three distinct vertices are required. Fewer points do not define a polygon with nonzero area.
• Consistent units: All coordinates must use the same unit (for example, all meters or all feet). Mixing units will distort the area.
• Flat, 2D geometry: The formula assumes a flat two-dimensional plane. It is not designed for spherical or highly curved surfaces.
• Numerical stability: Extremely large or extremely small coordinate values can lead to rounding errors in any numerical calculation; stick to reasonable magnitudes when possible.

If your shape is self-intersecting, you would need a more advanced algorithm that explicitly handles intersections and may interpret area in a different way (for example, counting some regions positively and some negatively).

Frequently Asked Questions

Can this calculator handle concave polygons?

Yes. Concave polygons are supported as long as the edges do not cross and the vertices are listed around the boundary in order. The shoelace formula does not require the polygon to be convex.

What happens if I enter the vertices out of order?

If the points are not in proper boundary order, the computed area will usually be wrong. There is no simple automatic fix, so take care to order your points as they appear around the polygon.

Does orientation (clockwise vs counterclockwise) matter?

Orientation matters for the internal sign of the area but not for the final result shown by this tool. The calculator always reports the absolute value, so you can enter coordinates in either direction.

What units does the calculator use?

The calculator is unit-agnostic. If your coordinates are in meters, the result is in square meters; if they are in feet, the result is in square feet. Always interpret the output in terms of the units you used for coordinates.

Can I use latitude and longitude coordinates?

Raw latitude and longitude are angles on a sphere, not planar coordinates. For small regions, you might approximate them as planar after projecting to a local coordinate system, but for accurate geospatial work you should convert to an appropriate map projection or use a dedicated GIS tool that accounts for Earth’s curvature.