Area of a Triangle With Coordinates Calculator

Sometimes you don't know the lengths of a triangle's sides or its angles. You only know the positions of its three vertices, each with a set of coordinates. If you enter those three points into this calculator, it will give you not only the area but also the perimeter, side lengths, interior angles and centroid.

At the core is the "shoe-string formula". This is a simple method of analytic geometry that allows you to convert six numbers directly into area without having to measure base or height.

Formula for area of triangle using coordinates:

Let's call the three vertices A, B and C and give each vertex an x-coordinate and a y-coordinate. The area of the triangle thus formed is as follows:

Here A is the point (x1, y1), B is the point (x2, y2) and C is the point (x3, y3). The vertical bars indicate absolute value so that a positive area results regardless of the order in which the points are arranged.

The same result can be obtained as a determinant; this is the origin of the name of the method. If one writes the coordinates in a grid and multiplies along the diagonal, it looks like shoelaces crossing.

A step by step method to calculate the area of a triangle given its coordinates:

The 'tie shoe' formula can be calculated manually in several steps.

First, calculate the expression inside of the vertical bars: multiply x1 by (y2-y3), x2 by (y3-y1), and then add that result to x3 multiplied by (y1-y2).

Then take the absolute value of this number, which removes all negative signs.

Finally divide the result by 2. The resulting value is the area of the triangle, with units being the square unit of the unit used in the coordinates.

Side lengths and perimeter.

Each side of the triangle is the straight line distance between two points and can be found with the Distance Formula. For example, let's find the length between point A and point B.

The same calculations are performed for the sides B to C and C to A, and when you add up the lengths of all three sides, you get the perimeter. This calculator calculates each side individually then sums them together.

Angle, centroid, shape

If the lengths of the three sides are known, any interior angle can be calculated using the law of cosines. The angle at vertex A is opposite side BC.

The centroid is the center of mass of a triangle and also the intersection point of its three medians. The calculation is even easier: it's the average of the coordinates of the three vertices.

Based on the side lengths this calculator will determine if the triangle is scalene, isosceles or equilateral. Also based on angles it will be classified as acute, right or obtuse.

Cases where points are collinear and area is zero:

The shoelace formula can also be used to quickly determine if three points are on a straight line. If the expression inside of the absolute value bars is exactly zero, then the three points lie on a line and do not enclose any area, so they do not form a triangle. All other values indicate that these points do form a real triangle. This calculator shows the case where the result is zero.

Before computing the absolute value, the sign of the expression also contains information. A positive value indicates that points A, B and C are oriented counterclockwise, while a negative value indicates clockwise orientation. This signed area is the result which the shoelace formula actually computes; in computer graphics, it can be used to determine which way a shape faces.

Example calculation:

Consider a triangle with vertices A(1,2), B(-1,1) and C(0,5). Applying the shoelace formula, the expression inside the absolute value brackets is the product of 1 (multiplied by the value of 1 minus 5), -1 (multiplied by the value of 5 minus 2) and 0 (multiplied by the value of 2 minus 1). So the result is -7, with 3 subtracted from -4 and 0 added. The absolute value is 7, and half of that gives an area of 3.5 square units.

The table below shows more triangles and their respective areas.

Fields of application for this method:

Surveyors and GIS analysts often use the triangle area formula because land is usually recorded as a list of vertex coordinates rather than a list of side lengths or angles. By splitting the land into triangles and adding up their areas, the total area can be found.

This formula also appears in computer graphics and game engines. Triangles are the basic building blocks of all 3D models, and the sign of the area determines whether a triangle faces towards or away from the camera. Students learn this formula in analytic geometry classes, along with other formulas such as the distance formula and the interior point formula. They also use these tools to check their calculations.

Tips for accurate results:

Make sure that the same unit is used for each coordinate. In this calculator you can change the units for each field but area will only make sense if x and y values use the same scale. If you are measuring points in meters, all fields should be in meters with the area being calculated in square meters.

Since the absolute value is used, the area does not change based on the order of the vertices, so A, B and C can be entered in any order. If you are interested in orientation, arrange the vertices to correspond with your desired view of the triangle and check the sign of the area.

This tool is for general education and everyday planning purposes. For surveying work, construction projects or other high risk activities these values must be checked against the project standards and tolerances.