Area from coordinates
The area of a polygon whose vertices are listed in order can be found with the shoelace formula. List the vertices in cyclic order, repeat the first at the end, then cross-multiply.
Key formula
Triangle A(x₁, y₁), B(x₂, y₂), C(x₃, y₃):
Area = ½ |x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|
Equivalent: ½ |x₁y₂ − x₂y₁ + x₂y₃ − x₃y₂ + x₃y₁ − x₁y₃|
Using the formula
Take the absolute value so the area is positive. If the result is 0, the points are collinear (they lie on one straight line). The name shoelace comes from the criss-cross pattern of the multiplications, like lacing a shoe.
For a polygon with more than three vertices, extend the pattern: multiply each x by the next y, subtract each y times the next x, sum all the terms, then halve the absolute value. The vertices must be taken in order around the boundary; jumping between non-adjacent corners gives a wrong answer. A quick way to handle a right-angled triangle or one with a horizontal or vertical side is to use ½ × base × height, which agrees with the shoelace result. Collinearity is a bonus use of the formula: if three points give zero area they lie on one line, a fast test that avoids finding gradients. Always label your vertices clearly before substituting to avoid sign slips.
Worked example
Triangle A(1, 1), B(5, 2), C(3, 6). Area = ½ |1(2 − 6) + 5(6 − 1) + 3(1 − 2)| = ½ |1(−4) + 5(5) + 3(−1)| = ½ |−4 + 25 − 3| = ½ (18) = 9 unit².
Remember
- Keep the vertices in cyclic order all the way round the shape.
- Always take the modulus for a positive area.
- Area = 0 means the points are collinear.