Shoelace Formula

The Key Result

Why It Works

1. 1
   List the vertices $(x_1,y_1), (x_2,y_2), \dots, (x_n,y_n)$ in order around the polygon, and treat $(x_{n+1}, y_{n+1}) = (x_1, y_1)$ to close the loop.
2. 2
   Each term $\tfrac12(x_i y_{i+1} - x_{i+1} y_i)$ is the signed area of the triangle formed by the origin and edge $i$ — it is the cross product $\tfrac12(\vec{P_i}\times\vec{P_{i+1}})$.
3. 3
   Summing these signed triangle areas, the contributions outside the polygon cancel and the inside is counted once, giving the polygon's signed area.
4. 4
   The sign depends on orientation (counterclockwise is positive), so take the absolute value to get the geometric area.

Worked Examples

Going Deeper

• Generalization: the shoelace sum is a sum of $2\times 2$ determinants, where each term $x_i y_{i+1} - x_{i+1} y_i = \det\!\begin{pmatrix} x_i & x_{i+1} \\ y_i & y_{i+1} \end{pmatrix}$ — the signed-area form. Dropping the absolute value keeps the orientation (counterclockwise $> 0$), and the same idea is the discrete face of Green's theorem and underlies Pick's theorem on lattice polygons.
• Where it appears: shoelace is the brute-force finisher for any AMC/AIME problem once coordinates are on the table — polygon areas, 'find the missing coordinate' problems (as in the Olympiad example), and a collinearity test (three points are collinear exactly when their shoelace area is $0$).
• Pitfall: the vertices MUST be listed in consecutive order around the boundary (all clockwise or all counterclockwise) — feeding them in a scrambled order computes the area of a different, self-intersecting figure and gives a wrong (often too-small) value. Also remember the final $\tfrac12$ and the absolute value; forgetting either is the most common slip.

Spot the Signal

• Use it when you have the coordinates of every vertex and want the enclosed area.
• You can describe the hard part as computing a polygon's area from its vertex coordinates, but a direct attack starts producing clutter.
• The problem rewards preserving structure instead of expanding, listing, or guessing too early.

Learn the Move

• Start by listing the coordinates in order and applying the cross-multiplied sum.
• Name the relation that makes Shoelace Formula legal before doing computation.
• Use the new relation to replace the messiest part of the problem with a cleaner one.
• Translate the result back to the quantity the problem actually asks for.

Avoid These Traps

• Listing the vertices out of order, or forgetting the absolute value and the factor of $\frac{1}{2}$.
• Applying Shoelace Formula because it sounds relevant, without checking the trigger first.
• Stopping after spotting the technique instead of finishing the calculation or proof.

MathGrit Coach Note