Overview

Shoelace is the fastest way to compute polygon area from coordinates.

Key Ideas

• For vertices $(x_i,y_i)$ in order, $A = \frac{1}{2}\left|\sum (x_i y_{i+1} - x_{i+1} y_i)\right|$.
• Order matters; list vertices clockwise or counterclockwise.

Core Skills

Order the Vertices

List points around the polygon (clockwise or counterclockwise). If the order is scrambled, reorder before applying the formula.

Use the Repeat-First Trick

Rewrite the first vertex at the end to make the diagonal products easier.

Take Absolute Value Last

Compute the signed area first, then take the absolute value at the end.

Worked Example

Triangle has vertices $(0,0)$, $(12,0)$, $(0,5)$. Find the area.

$A = \frac{1}{2}|0(0-5) + 12(5-0) + 0(0-0)| = 30$.

More Examples

Example 1: Quadrilateral

Find the area of the polygon $(1,1)$, $(4,1)$, $(4,3)$, $(1,3)$.

This is a rectangle of area $6$, which matches shoelace.

Example 2: Reordered Points

Why does $(0,0)$, $(2,2)$, $(2,0)$ give the wrong area if used in that order?

The points are not listed around the triangle. Reorder as $(0,0)$, $(2,0)$, $(2,2)$.

Strategy Checklist

• Put vertices in cyclic order.
• Repeat the first vertex at the end.
• Apply absolute value at the end.

Common Pitfalls

• Listing vertices out of order.
• Forgetting the absolute value.
• Mixing polygon order (clockwise and counterclockwise).

Practice Problems