The Shoelace Formula for Area

Ever get stuck finding a triangle's area when you only have its corner points (coordinates)? Forget complicated base-height calculations! Here's a trick!

Subject: geometry • Classes: 6–12 • Difficulty: intermediate (9-10)

The Trick

The 'Shoelace Formula' (or Surveyor's Formula) quickly finds the area of any polygon given its vertices. For a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$: 1. List the coordinates vertically, repeating the first coordinate at the end: $x_1 \quad y_1$ $x_2 \quad y_2$ $x_3 \quad y_3$ $x_1 \quad y_1$ 2. Calculate 'Sum A': Multiply diagonally downwards (left $x$ to right $y$) and add them: $(x_1y_2 + x_2y_3 + x_3y_1)$. 3. Calculate 'Sum B': Multiply diagonally upwards (right $y$ to left $x$) and add them: $(y_1x_2 + y_2x_3 + y_3x_1)$. 4. The Area is $\frac{1}{2} |\text{Sum A} - \text{Sum B}|$. WHY it works: This formula cleverly sums the signed areas of trapezoids formed by projecting the polygon's edges onto the x-axis, with the cross-multiplication method simplifying the calculation.

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