Area of a Convex Polygon: Shoelace Formula & Examples

Area of a Convex Polygon

The coordinates (x₁, y₁), (x₂, y₂), (x₃, y₃), . . . , (x_n, y_n) of a convex polygon are arranged in the "determinant" below. The coordinates must be taken in counterclockwise order around the polygon, beginning and ending at the same point.

Convex polygon divided into triangles to show how its area can be found by adding triangle areas.

Key Formula

A = \frac{1}{2} \left| \sum_{i=1}^{n} \left( x_i \, y_{i+1} - x_{i+1} \, y_i \right) \right|

Where:

$A$ = Area of the convex polygon
$n$ = Number of vertices of the polygon
$(x_i, y_i)$ = Coordinates of the i-th vertex, listed in counterclockwise order
$(x_{n+1}, y_{n+1})$ = Equal to (x_1, y_1) — the first vertex is repeated at the end to close the polygon

Worked Example

Problem: Find the area of the convex quadrilateral with vertices A(1, 1), B(5, 1), C(6, 4), and D(2, 5), listed in counterclockwise order.

Step 1: List the vertices in counterclockwise order and repeat the first vertex at the end to close the polygon.

(1,1),\;(5,1),\;(6,4),\;(2,5),\;(1,1)

Step 2: Compute each "forward" cross-product term: multiply each x-coordinate by the y-coordinate of the next vertex.

x_1 y_2 + x_2 y_3 + x_3 y_4 + x_4 y_1 = (1)(1) + (5)(4) + (6)(5) + (2)(1) = 1 + 20 + 30 + 2 = 53

Step 3: Compute each "backward" cross-product term: multiply each y-coordinate by the x-coordinate of the next vertex.

y_1 x_2 + y_2 x_3 + y_3 x_4 + y_4 x_1 = (1)(5) + (1)(6) + (4)(2) + (5)(1) = 5 + 6 + 8 + 5 = 24

Step 4: Subtract the backward sum from the forward sum and take half the absolute value.

A = \frac{1}{2}|53 - 24| = \frac{1}{2}(29) = 14.5

Answer: The area of the quadrilateral is 14.5 square units.

Another Example

This example uses a five-sided polygon instead of four sides, showing that the Shoelace Formula works for any number of vertices, not just triangles or quadrilaterals.

Problem: Find the area of the convex pentagon with vertices (0, 0), (4, 0), (6, 3), (3, 6), and (0, 4), listed in counterclockwise order.

Step 1: Write the vertices in order and repeat the first vertex at the end.

(0,0),\;(4,0),\;(6,3),\;(3,6),\;(0,4),\;(0,0)

Step 2: Compute the forward products (x_i × y_{i+1}).

(0)(0) + (4)(3) + (6)(6) + (3)(4) + (0)(0) = 0 + 12 + 36 + 12 + 0 = 60

Step 3: Compute the backward products (y_i × x_{i+1}).

(0)(4) + (0)(6) + (3)(3) + (6)(0) + (4)(0) = 0 + 0 + 9 + 0 + 0 = 9

Step 4: Apply the formula to find the area.

A = \frac{1}{2}|60 - 9| = \frac{1}{2}(51) = 25.5

Answer: The area of the pentagon is 25.5 square units.

Frequently Asked Questions

Does the Shoelace Formula work for concave (non-convex) polygons?

Yes. The Shoelace Formula actually works for any simple polygon (one that does not cross itself), whether convex or concave. The key requirement is that the vertices are listed in order around the boundary and the polygon's sides do not intersect each other. The absolute value in the formula ensures a positive result regardless of whether the vertices are listed clockwise or counterclockwise.

What happens if I list the vertices in clockwise order instead of counterclockwise?

If you list the vertices clockwise, the sum inside the formula will be negative instead of positive. Because the formula takes the absolute value, you still get the correct area. However, some textbooks present the formula without the absolute value and explicitly require counterclockwise order to guarantee a positive result.

How do I know if the vertices are in counterclockwise order?

Plot the points on a coordinate plane and trace from one vertex to the next. If you travel around the polygon in a direction that keeps the interior on your left, you are going counterclockwise. Alternatively, compute the signed area without the absolute value — a positive result means counterclockwise, and a negative result means clockwise.

Shoelace Formula (Coordinate Method) vs. Area of a Regular Polygon

	Shoelace Formula (Coordinate Method)	Area of a Regular Polygon
Input needed	Coordinates of each vertex	Side length (or apothem) and number of sides
Formula	A = ½\|Σ(x_i · y_{i+1} − x_{i+1} · y_i)\|	A = ½ · perimeter · apothem
Shape requirement	Any simple polygon (convex or concave)	Only regular (equilateral and equiangular) polygons
When to use	When vertex coordinates are given on a coordinate plane	When the polygon is regular and you know side length or apothem

Why It Matters

This formula appears frequently in coordinate geometry courses and standardized tests whenever you are given polygon vertices on a grid. It is also foundational in computer graphics and computational geometry, where programs calculate the areas of thousands of polygons per second using this exact method. Mastering it prepares you for analytic geometry proofs and for working with vectors and determinants in later courses.

Common Mistakes

Mistake: Forgetting to repeat the first vertex at the end of the list.

Correction: The formula requires a "closed loop." After listing all n vertices, write the first vertex again as the (n+1)-th entry so the last cross-product term connects the final vertex back to the starting vertex.

Mistake: Listing vertices in a random order rather than sequentially around the perimeter.

Correction: The vertices must be ordered as you walk around the boundary of the polygon (either all counterclockwise or all clockwise). If they are jumbled, the formula will compute the wrong area or the area of a self-intersecting figure.

Related Terms

Coordinates — Vertex positions used in the formula
Convex — Shape property where all interior angles < 180°
Polygon — The general class of shapes this applies to
Determinant — Each cross-product term is a 2×2 determinant
Counterclockwise — Standard vertex ordering for the formula
Area of a Regular Polygon — Alternative area formula for equilateral polygons
Analytic Geometry — Branch of math using coordinates to solve geometry