Koordinaten-Vieleck-Rechner

The Polygon Coordinates Calculator uses the Shoelace formula (also called the surveyor's formula or Gauss's area formula) to find the area of any simple polygon — regular or irregular — from its (x, y) vertex coordinates alone. No side lengths, no angles, no height measurements needed. Just plug in the vertices in order. This tutorial explains the formula, why it works (in geometric terms), and walks through worked examples for both convex and concave polygons.

The Shoelace formula

For a polygon with n vertices listed in order as (x₁, y₁), (x₂, y₂), ..., (xₙ, yₙ):

Area = ½ × |Σ (xᵢ × yᵢ₊₁ − xᵢ₊₁ × yᵢ)|

The sum runs over all consecutive vertex pairs, treating (xₙ₊₁, yₙ₊₁) as the same as (x₁, y₁) (wrap around). The absolute value bars handle the case where vertices are listed clockwise (which gives negative under the sum); the result is always a positive area.

The "Shoelace" name

The formula gets its name from the way you compute it visually:

1. Write all the x-coordinates in one column, y-coordinates in another, REPEATING the first row at the bottom (to close the polygon).
2. Multiply down-and-right (each xᵢ times the y_{i+1}). Sum these.
3. Multiply down-and-LEFT (each yᵢ times the x_{i+1}). Sum these.
4. Take the absolute difference of the two sums, divide by 2.

The diagonal multiplication pattern looks like a shoelace zigzag — hence the name.

Worked example 1 — square via Shoelace

Square with vertices (0, 0), (4, 0), (4, 3), (0, 3) — actually a 4 × 3 rectangle.

Compute the down-right products and down-left products:

Down-right: (0×0) + (4×3) + (4×3) + (0×0) = 0 + 12 + 12 + 0 = 24

Wait, let me redo this with the correct pairing. The formula is Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ).

• (x₁y₂ − x₂y₁) = (0×0) − (4×0) = 0
• (x₂y₃ − x₃y₂) = (4×3) − (4×0) = 12
• (x₃y₄ − x₄y₃) = (4×3) − (0×3) = 12
• (x₄y₁ − x₁y₄) = (0×0) − (0×3) = 0

Sum = 0 + 12 + 12 + 0 = 24. Area = |24| / 2 = 12.

Verification: A 4 × 3 rectangle has area 12. ✓

Worked example 2 — triangle

Triangle with vertices (0, 0), (6, 0), (3, 4).

• (0×0 − 6×0) = 0
• (6×4 − 3×0) = 24
• (3×0 − 0×4) = 0

Sum = 24. Area = 24 / 2 = 12.

Verification: triangle base 6, height 4, area = ½ × 6 × 4 = 12. ✓

Worked example 3 — irregular pentagon

Pentagon with vertices (0, 0), (5, 0), (6, 3), (3, 5), (−1, 3).

• (0×0 − 5×0) = 0
• (5×3 − 6×0) = 15
• (6×5 − 3×3) = 21
• (3×3 − (−1)×5) = 9 + 5 = 14
• ((−1)×0 − 0×3) = 0

Sum = 50. Area = 50 / 2 = 25.

Notice: no need to compute side lengths or break the pentagon into triangles. Just the vertex coordinates.

Why does Shoelace work?

Intuition: the Shoelace formula computes the signed area swept out by going around the polygon. Each term (xᵢyᵢ₊₁ − xᵢ₊₁yᵢ) is twice the signed area of the triangle formed by the origin, vertex i, and vertex i+1. Summing all these gives twice the polygon area. Divide by 2 to recover the area itself.

The absolute value handles the case where vertices are listed clockwise (giving a negative result) versus counter-clockwise (positive). Both orderings give the same absolute area.

Counter-clockwise vs clockwise

Listing the vertices counter-clockwise gives a POSITIVE sum. Clockwise gives a NEGATIVE sum.

This is intentional — it lets the formula detect orientation. In some contexts (computational geometry, polygon winding), the sign tells you which "side" of the polygon you're tracing. For pure area calculation, just take the absolute value.

Important: vertices must be in order

The Shoelace formula requires vertices listed in order around the polygon boundary — either consistently clockwise or consistently counter-clockwise. Listing them out of order (jumping around) creates a self-intersecting "polygon" that doesn't exist physically, and the formula returns a different (smaller) area.

Concave polygons

Shoelace works for concave (non-convex) polygons too — as long as the polygon is simple (doesn't self-intersect). Just list the vertices in their natural boundary order.

Self-intersecting polygons

For polygons with crossing edges (like a star drawn with a single continuous line, or a "bowtie"), Shoelace returns a value that depends on the crossings — typically the "net" signed area where some regions count positive and others negative. For most practical purposes, this isn't the result you want; ensure your polygon is simple before using Shoelace.

Real-world applications

• Surveying. Computing land area from corner GPS coordinates. The Shoelace formula is exactly how property surveyors calculate plot areas.
• GIS / Mapping. Computing the area of a region defined by latitude-longitude polygon vertices (with a flat-Earth approximation for small regions).
• Computer graphics. Computing polygon areas for collision detection, rendering, or geometric algorithms.
• Architecture and design. Computing irregular floor plan areas from CAD coordinates.
• Mathematics — Pick's theorem. Counts the lattice points inside a polygon with integer vertices, related to the Shoelace area.

Common mistakes

• Forgetting to close the polygon. The last vertex must connect back to the first. Either include (xₙ₊₁, yₙ₊₁) = (x₁, y₁) explicitly or treat the wraparound implicitly.
• Listing vertices out of order. Random ordering creates a self-intersecting shape with wrong area. Always trace the boundary in order.
• Forgetting the ½. The Shoelace sum is TWICE the polygon area. Divide by 2 at the end.
• Forgetting the absolute value. The result can be negative (clockwise listing). Areas are always positive — take |result|.
• Using on self-intersecting polygons. Shoelace gives "signed area" for self-intersecting figures; this is not the same as physical area.