Area of a triangle from coordinates
area = ½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|.
Key formulas
Shoelace idea
List the vertices in order around the triangle (clockwise or anticlockwise). The absolute value keeps area positive.
Example
A(0,0), B(4,0), C(0,6) → area = ½|0+4(6−0)+0|=12.
Collinear test
If the formula returns 0, the points are collinear and do not form a triangle.
Worked example
Worked example
Problem. Find the area of the triangle with vertices (0,0), (4,0) and (0,6).
Solution. Area = ½|0(0−6) + 4(6−0) + 0(0−0)| = ½|24| = 12.
Order
Reversing order only changes the sign inside the absolute value.
Review quiz
Points (1,1), (5,1), (1,7). Area is:
Practice questions
- Points (1,1), (6,1), (3,5). Find the area.
- Show that (1,2), (3,6), (5,10) are collinear using area.
Study notes
Can I use vectors?
Yes — half the magnitude of AB × AC in 2D.