GeometryDifficulty 25-8036 tagged problems

Shoelace Formula

Shoelace Formula is the habit of computing a polygon's area from its vertex coordinates. In contest math, that habit turns a crowded setup into a relation the student can test, bound, count, or compute. MathGrit teaches it as a recognizable signal, a deliberate move, and a final translation back to the original question.

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The Key Result

$$A = \frac{1}{2}\left| \sum_{i=1}^{n} \bigl(x_i y_{i+1} - x_{i+1} y_i\bigr) \right| \qquad (\text{indices mod } n)$$

You compute the exact area of any polygon directly from the coordinates of its vertices, taken in order around the boundary.

Why It Works

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
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
Summing these signed triangle areas, the contributions outside the polygon cancel and the inside is counted once, giving the polygon's signed area.
4
The sign depends on orientation (counterclockwise is positive), so take the absolute value to get the geometric area.

Worked Examples

Example 1

Find the area of the quadrilateral with vertices $(1,1)$, $(4,1)$, $(5,4)$, $(2,3)$, listed in order around the boundary.

1
Form each cross term $x_i y_{i+1} - x_{i+1} y_i$ going around: $(1\cdot1 - 4\cdot1) = -3$, then $(4\cdot4 - 5\cdot1) = 11$, then $(5\cdot3 - 2\cdot4) = 7$, then $(2\cdot1 - 1\cdot3) = -1$.
2
Sum the terms: $-3 + 11 + 7 - 1 = 14$.
3
Apply $A = \tfrac12|{\sum}| = \tfrac12\,|14| = 7$.

Answer:

Area $= 7$

Warm-up

Find the area of the triangle with vertices $(0,0)$, $(5,1)$, $(2,6)$.

1
Shoelace cross terms $x_i y_{i+1} - x_{i+1} y_i$ around the triangle: $(0\cdot 1 - 5\cdot 0) = 0$, then $(5\cdot 6 - 2\cdot 1) = 28$, then $(2\cdot 0 - 0\cdot 6) = 0$.
2
Sum the terms: $0 + 28 + 0 = 28$.
3
Area $= \tfrac12|28| = 14$.

Answer:

Area $= 14$

Contest level

A pentagon has vertices $(0,0)$, $(4,0)$, $(5,3)$, $(2,5)$, $(-1,3)$ in order. Find its area.

1
Form the cross terms $x_i y_{i+1} - x_{i+1} y_i$ going around: $(0\cdot 0 - 4\cdot 0) = 0$, $(4\cdot 3 - 5\cdot 0) = 12$, $(5\cdot 5 - 2\cdot 3) = 19$, $(2\cdot 3 - (-1)\cdot 5) = 11$, $((-1)\cdot 0 - 0\cdot 3) = 0$.
2
Sum: $0 + 12 + 19 + 11 + 0 = 42$.
3
Area $= \tfrac12|42| = 21$.

Answer:

Area $= 21$

Olympiad / Challenge

The triangle with vertices $A = (0,0)$, $B = (6,2)$, $C = (t, 5)$ has area $13$. Find all possible values of $t$.

1
With one vertex at the origin the shoelace area is $\tfrac12\,|x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B)|$.
2
Substitute $A = (0,0)$, $B = (6,2)$, $C = (t,5)$: $\tfrac12\,|0\cdot(2 - 5) + 6\cdot(5 - 0) + t\cdot(0 - 2)| = \tfrac12\,|30 - 2t|$.
3
Set equal to $13$: $|30 - 2t| = 26$, so $30 - 2t = 26$ or $30 - 2t = -26$.
4
Solve both: $t = 2$ or $t = 28$. (The two values are the two positions of $C$ — one on each side — that give the same area $13$.)

Answer:

$t = 2$ or $t = 28$.

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

Go around the polygon once in order; cross-multiply, halve, take the size.

Try it on:

Find the area of a polygon given its vertex coordinates.