Area by Coordinates (Shoelace)

Key idea

Area of a polygon from vertices.

In depth

List the vertices in order and compute A = ½|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)|. It sums signed areas of triangles from the origin and works for any simple polygon.

Worked example

Triangle (0,0), (4,0), (0,3): ½|0 + 12 + 0 − (0 + 0 + 0)| = 6.

Formula

Real-world uses

• Calculating the area of an irregular plot from its corner coordinates in land surveying

Common mistake

Listing vertices out of order, which gives wrong areas.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Geometry

• Power of a Point – For a point and a circle, products of segment lengths along secants are equal.
• Cyclic Quadrilaterals – Opposite angles of a quadrilateral inscribed in a circle sum to 180°.
• Cavalieri’s Principle – Solids with equal cross-sectional areas at every height have equal volume.
• Parabola (Geometric Definition) – Points equidistant from a focus and a directrix.
• Ellipse – Sum of distances to two foci is constant.
• Hyperbola – Difference of distances to two foci is constant.