How to use
- Choose the number of vertices, then enter X and Y separately for each row, following the boundary in order. No separators are needed.
- Choose one length unit for all coordinates. Negative coordinates and decimal numbers are allowed; use a point or comma decimal separator, without thousands separators.
- Calculate, then check the numbered outline before using its area and perimeter. The last point is connected to the first automatically.
Changing the number of vertices adds or removes rows at the end. Values in temporarily removed rows are retained until Clear. Examples below list the vertices in boundary order.
The drawing updates as soon as the inputs describe valid geometry. If a value is missing or invalid, the drawing stays hidden until the inputs are ready. Editing clears the previous numerical result and disables Copy result. Press Calculate to confirm the current numbers, then copy them as text. Clear empties the numerical inputs while keeping your selections.
Formula and interpretation
The shoelace formula adds signed contributions from consecutive vertex pairs, including the closing edge. Taking the absolute value gives the enclosed area. Perimeter is the sum of straight-line distances between consecutive points. A concave polygon is valid: its inward corner is preserved rather than replaced by an outer hull.
A = ½ |Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|
L = Σ √((xᵢ₊₁ − xᵢ)² + (yᵢ₊₁ − yᵢ)²)
(xₙ₊₁, yₙ₊₁) = (x₁, y₁)
Worked example
Rectangle: 0 0; 4 0; 4 3; 0 3 gives area 12 square units and perimeter 14 units. With metres selected, these are 12 m² and 14 m. Reversing all vertices gives the same measurements.
Triangle: 0 0; 4 0; 0 3 gives area 6 square units and perimeter 12 units. Its sides are 4, 5 and 3; the closing edge is included even without repeating 0 0.
Concave shape: 0 0; 4 0; 4 3; 2 1; 0 3 has area 8 square units and perimeter 10 + 4√2 ≈ 15.66 units. The inward vertex removes a triangle of area 4 from the 4-by-3 rectangle.
Uses and limits
Use this for coordinate geometry, a flat cutout outline or a simple plan with known Cartesian coordinates. It does not infer missing corners, reorder points or calculate a shape from side lengths alone. Holes, disconnected boundaries, curves, geographical latitude/longitude, maps, CAD and file uploads are not supported. Enter one simple boundary of 3–30 vertices.
Calculations use floating-point arithmetic and translated, scaled coordinates to reduce cancellation. Ordinary results show up to two decimal places; small nonzero values use scientific notation. Coordinates are limited to zero or magnitudes from 10⁻⁹ to 10⁹, and results to 10¹⁵. Edges shorter than 10⁻¹⁰ of the overall span and almost-zero areas or almost-touching boundaries may be rejected because their geometry is numerically ambiguous.
Frequently asked questions
Should I repeat the first vertex at the end?
You may, but it is unnecessary. One matching closing vertex is removed before counting and measuring. Repeated vertices elsewhere are rejected.
Can I enter clockwise or counterclockwise points?
Both work. Reverse the entire order to change direction without changing the boundary. Sorting only some points can create a different polygon or crossing.
Why is a bow-tie outline rejected?
A crossing boundary does not have the single simple interior used by this calculator. Check the order. The tool never guesses a replacement outline.
Are negative coordinates a problem?
No. Shifting every point by the same x and y offsets leaves area and perimeter unchanged. The drawing keeps equal horizontal and vertical scales, so a rectangle stays proportional.