Enter the three (x, y) coordinates of a triangle's vertices to calculate its area using the Shoelace formula. Works with any real numbers including negative coordinates and decimals.
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Standard formula
Shoelace (coordinate geometry) method. Exact for all real-number inputs.
How to Find the Area of a Triangle from Coordinates
When you know the Cartesian coordinates of a triangle's three vertices, you do not need to know the base or height directly. The Shoelace formula (also known as the coordinate method or surveyor's formula) gives the exact area using only the (x, y) values of the three points.
The Shoelace Formula
For a triangle with vertices A(x1, y1), B(x2, y2), and C(x3, y3):
Area = 0.5 x |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
The vertical bars | | denote the absolute value, which ensures a positive area regardless of whether the vertices are listed clockwise or anticlockwise.
Worked Example
Using the default values: A(0, 0), B(6, 0), C(3, 4).
Step
Calculation
Result
x1(y2 - y3)
0 x (0 - 4)
0
x2(y3 - y1)
6 x (4 - 0)
24
x3(y1 - y2)
3 x (0 - 0)
0
Sum
|0 + 24 + 0|
24
Area
0.5 x 24
12 m²
This matches what we can verify independently: the base AB has length 6 m and the height from C perpendicular to AB is 4 m, so Area = 0.5 x 6 x 4 = 12 m².
When Do Coordinates Give a Zero Area?
If the three points are collinear (all lie on the same straight line), the Shoelace formula returns zero. This means no triangle can be formed from those points. The calculator flags this as a degenerate case.
Perimeter from Coordinates
Once you have the three vertices, you can also compute the side lengths using the distance formula and add them together:
Side AB = sqrt((x2 - x1)² + (y2 - y1)²)
Side BC = sqrt((x3 - x2)² + (y3 - y2)²)
Side CA = sqrt((x1 - x3)² + (y1 - y3)²)
Perimeter = AB + BC + CA
This calculator computes all three side lengths automatically and classifies the triangle as equilateral, isosceles, or scalene based on the side lengths.
Method: Shoelace formula (coordinate geometry). Also known as the surveyor's formula. Exact for all real-number coordinate inputs. Perimeter uses the Euclidean distance formula. Triangle classification uses side-length ratios rounded to 6 significant figures.
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