What formula is used to find the area of a triangle from coordinates?

The Shoelace formula (also called the coordinate geometry method or cross-product method) is used. For a triangle with vertices (x1, y1), (x2, y2), and (x3, y3), the area is: Area = 0.5 x |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|. The absolute value ensures a positive result regardless of the order the vertices are listed.

Does it matter what order I enter the coordinates?

No. The Shoelace formula produces the same area whether you enter the vertices in clockwise or anticlockwise order because the absolute value is taken at the end. However, if all three points are collinear (i.e., they lie on the same straight line), the formula returns zero because no triangle can be formed.

Can coordinates be negative or decimal?

Yes. The formula works with any real numbers, including negative coordinates and decimals. For example, a triangle with vertices at (-3, 1), (2, -4), and (5, 3) is handled correctly. The calculator accepts any numeric input including negative values and fractional coordinates.

Area of a Triangle with Coordinates Calculator

This calculator finds the exact area of a triangle when you know the Cartesian coordinates of its three corners, rather than its base and height. It uses the Shoelace formula (also called the coordinate geometry or surveyor's formula), a standard method in maths, surveying, mapping and computer graphics for working out area straight from a set of x, y points. You enter the x and y values for vertex A, vertex B and vertex C, choosing decimals or negative numbers if needed, and can optionally pick a unit (cm, m, km, mm, in or ft) so the result is labelled correctly. The calculator returns the triangle's area, its perimeter (calculated from the three side lengths using the distance formula), and classifies the triangle as equilateral, isosceles or scalene based on those side lengths. It also shows the full step-by-step working, from each term in the formula through to the final sum and area, so you can check the calculation or use it for homework or professional reference. It does not matter whether you list the vertices clockwise or anticlockwise, since the formula always returns a positive area. If the three points sit on the same straight line, the calculator returns zero and flags the triangle as degenerate, meaning no real triangle can be formed from those coordinates.

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Standard formula Shoelace (coordinate geometry) method. Exact for all real-number inputs.

Vertex A

Point A (x₁, y₁)

x₁

y₁

Vertex B

Point B (x₂, y₂)

x₂

y₂

Vertex C

Point C (x₃, y₃)

x₃

y₃

Units (optional label)

Triangle Area

Area
12
m²

Perimeter

Triangle type

Isosceles

by side lengths

Step-by-Step Working

FormulaArea = 0.5 x |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Vertices enteredA(0, 0), B(6, 0), C(3, 4)

Step 1: x1(y2 - y3)0 x (0 - 4) = 0

Step 2: x2(y3 - y1)6 x (4 - 0) = 24

Step 3: x3(y1 - y2)3 x (0 - 0) = 0

Sum inside | ||0 + 24 + 0| = 24

Area = 0.5 x 24= 12 m²

Result: The triangle has an area of 12 m².

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.

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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.