The area of a polygon can be found from its vertices by multiplying coordinates in a fixed pattern. The pattern only works when the points are listed in order around the shape.
This lesson is part of coordinate geometry. It pairs with equations of parallel and perpendicular lines.
What are the steps of the shoelace method?
- Sketch the shape and list the vertices in order, then repeat the first vertex at the end.
- Add the products of each x with the next y.
- Add the products of each y with the next x.
- Area = ½ × |first sum − second sum|.
Worked example 1: a triangle
The vertices are A(1, 1), B(7, 2) and C(4, 6).
First sum: 1 × 2 + 7 × 6 + 4 × 1 = 2 + 42 + 4 = 48.
Second sum: 1 × 7 + 2 × 4 + 6 × 1 = 7 + 8 + 6 = 21.
Area = ½ × |48 − 21| = 13.5 square units.
Check with a second route. AB = (6, 1) and AC = (3, 5), and 6 × 5 − 1 × 3 = 27, so the area is 27 ÷ 2 = 13.5.
Worked example 2: a quadrilateral
The vertices in order are P(0, 0), Q(6, 0), R(8, 5) and S(2, 4).
First sum: 0 × 0 + 6 × 5 + 8 × 4 + 2 × 0 = 62. Second sum: 0 × 6 + 0 × 8 + 5 × 2 + 4 × 0 = 10.
Area = ½ × |62 − 10| = 26 square units.
Check by splitting along PR: triangle PQR = ½ × 6 × 5 = 15, and triangle PRS = ½ × |8 × 4 − 5 × 2| = 11. Together, 15 + 11 = 26.
The mistake that costs marks
A common slip is to list the points in a crossing order. Take the same four points as P, R, Q, S.
| Order | First sum | Second sum | Area |
|---|---|---|---|
| P, Q, R, S (around the shape) | 62 | 10 | 26 |
| P, R, Q, S (crossing) | 24 | 30 | 3 |
The second order joins P to R and Q to S across the shape, so the formula measures a twisted figure. A quick sketch shows which order follows the boundary.
Check yourself
Find the area of the triangle with vertices (−2, 1), (4, −1) and (2, 5).
Answer
First sum: (−2)(−1) + 4 × 5 + 2 × 1 = 2 + 20 + 2 = 24.
Second sum: 1 × 4 + (−1)(2) + 5 × (−2) = 4 − 2 − 10 = −8.
Area = ½ × |24 − (−8)| = ½ × 32 = 16 square units.
What to study next
Test yourself with the coordinate geometry practice set. When a problem can be solved by more than one route, see geometry problems with several possible approaches.
The word-problem structure worksheet helps you list the points, and a mistake log shows whether the order is your slip. To work with a teacher, see online one-to-one Additional Mathematics tuition.