Area from Coordinates: Ordered Vertices and Shoelace

List vertices in boundary order

Write the polygon’s vertices clockwise or counterclockwise around its boundary, then repeat the first vertex at the end of the list. Boundary order is essential. Sorting by coordinate size or jumping across the figure can create a self-intersecting path that describes a different region. For a simple polygon, the shoelace formula gives A = |Σ(xi yi+1 − yi xi+1)|/2. The absolute value removes the sign introduced by choosing clockwise rather than counterclockwise orientation.

Calculate one complete example

Take A = (1,1), B = (5,1), C = (4,4) and D = (1,3), in that order. The forward products are 1×1, 5×4, 4×3 and 1×1, summing to 34. The backward products are 1×5, 1×4, 4×1 and 3×1, summing to 16. Area is |34 − 16|/2 = 9 square coordinate units. The closing products involving D and A are part of the formula; omitting them can change the result whenever they do not cancel.

Check by splitting into triangles

Diagonal AC divides the example into triangles ABC and ACD. Triangle ABC has base AB = 4 and perpendicular height 3, so its area is 6. Triangle ACD has vertical base AD = 2 and horizontal distance 3 from C to that base, so its area is 3. The total is 9, agreeing with the coordinate products. Reversing the vertex order changes the signed sum from 18 to −18 but leaves the final absolute area unchanged.

Recognize what the formula does not infer

For coordinates in metres, the result is square metres. For a scaled drawing, apply the squared scale factor to the area. Latitude and longitude require an appropriate geographic or projected method before ordinary plane-area arithmetic is used. A self-crossing bow-tie path can cause signed regions to cancel; taking the absolute value at the end does not recover the total of all visible lobes. Resolve the intended simple boundaries first, and calculate separate regions if the drawing includes holes or disconnected pieces.

Check whether the polygon is regular

A polygon has a closed boundary made from straight segments. A regular polygon has equal side lengths and equal interior angles. Equal sides alone do not guarantee regularity: a rhombus can have equal sides but unequal angles. Likewise, a rectangle can have equal angles without equal sides. The regular-polygon calculator takes one side length and a side count, so its area and angle results depend on the regularity assumption. Do not feed an average side length into that form and treat the result as the measured area of an irregular plot.

Keep the measurement types distinct

Side lengths and perimeter use ordinary length units; area uses square units; angles use degrees or radians. If every length in a similar figure is multiplied by k, perimeter multiplies by k and area by k², while angles remain unchanged. This gives a direct check on a calculation with a converted side length. Converting centimetres to metres before calculating should describe the same figure as converting the final square centimetres to square metres, provided the area conversion factor is squared rather than copied from the length conversion.

List enough information to determine the answer

A side count identifies a broad class of shapes, but it does not usually determine their size or area. A perimeter likewise allows many different irregular shapes. Label which conditions are given and which are assumptions, and sketch the boundary before substituting numbers. For coordinate data, preserve the order around the boundary; sorting vertices simply by x-coordinate can create crossing segments. A photograph or drawing is not automatically to scale. Calculator output describes the ideal geometry entered into its fields, so retain the underlying measurements and assumptions when reporting a result.

Related guides

Coordinate Geometry: Distance, Midpoint and Slope

Use a coordinate geometry formula sheet with one checked example, distinguish distance from midpoint and slope, and recognize vertical-line exceptions.

Polygon Area Formulas for Regular and Irregular Shapes

Find regular-polygon area from perimeter and apothem, derive a side-length formula, and choose decomposition for an irregular boundary.

Area Formulas for Shapes and Composite Regions

Choose area formulas for common plane shapes, use a supported multi-shape input, and handle overlaps and cut-outs explicitly.