Area of Triangle with Coordinates Calculator
Enter the x and y coordinates of three vertices and get the triangle's area (via the shoelace formula), perimeter, all three side lengths, all three interior angles, and the centroid. A step-by-step panel shows the full working so you can follow each calculation.
Formula
Worked example
Vertices A(0,0), B(4,0), C(2,3): Area = 0.5*|0*(0-3)+4*(3-0)+2*(0-0)| = 0.5*|0+12+0| = 6 sq units.
The shoelace formula explained
The shoelace formula (also called the surveyor's formula or Gauss's area formula) computes the area of any polygon from the coordinates of its vertices. For a triangle with vertices A(x1, y1), B(x2, y2), and C(x3, y3), it reads: Area = 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)| The expression inside the absolute value is the signed area of the triangle: positive when the vertices are listed counter-clockwise, negative when they are listed clockwise. Taking the absolute value and halving gives the true area regardless of vertex order. The name "shoelace" comes from a visual pattern: if you lay out the coordinates in two columns and cross-multiply adjacent pairs, the differences between the diagonal products trace a lace-like path.
Side lengths and angles from coordinates
Each side length is the straight-line distance between two vertices, computed with the distance formula: length = sqrt((x2 - x1)^2 + (y2 - y1)^2). The three interior angles are found using the dot product: at each vertex, form two vectors that point toward the other two vertices, compute their dot product and magnitudes, then apply the arccosine. The three angles always sum to exactly 180 degrees. The centroid - the balance point of the triangle - is simply the average of the three x-coordinates paired with the average of the three y-coordinates: G = ((x1+x2+x3)/3, (y1+y2+y3)/3). It is the point where the three medians of the triangle intersect.
Collinearity and degenerate triangles
When the signed area in the shoelace formula equals zero, all three points lie on the same straight line - they are collinear - and no real triangle is formed. This is called a degenerate triangle. A quick collinearity test is therefore: check whether x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) = 0. If it does, the "triangle" is actually a line segment or a single point. This condition is also used in computational geometry to determine orientation (clockwise vs. counter-clockwise) of a sequence of points.
Practical uses
Finding the area from coordinates arises in many real-world contexts: surveyors use it to compute land parcel areas from GPS waypoints, computer graphics engines use signed area to determine polygon winding order, GIS software uses it to calculate areas on a map plane, and physics simulations use it to check whether a point falls inside a triangle (a point P is inside triangle ABC if the sub-triangles PAB, PBC, and PCA have areas that sum to the area of ABC). The shoelace formula generalises directly to any polygon with n vertices, making it one of the most widely used area formulas in applied mathematics.
Triangle types by angles
| Type | Angle condition | Notes |
|---|---|---|
| Acute | All angles < 90 deg | All vertices "point inward" |
| Right | One angle = 90 deg | Pythagorean theorem applies |
| Obtuse | One angle > 90 deg | One vertex "points outward" |
| Equilateral | All angles = 60 deg | All sides equal |
| Isosceles | Two angles equal | Two sides equal |
| Scalene | All angles different | All sides different |
| Degenerate | Area = 0 | Points are collinear |
Classification based on the three interior angles.
Frequently asked questions
What formula does this calculator use?
It uses the shoelace formula: Area = 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|. This formula works for any triangle regardless of its orientation or position in the coordinate plane, and it gives zero (correctly) when the three points are collinear.
Does the order of the vertices matter?
The order does not affect the area because the formula takes the absolute value of the signed area. Listing the vertices clockwise versus counter-clockwise only changes the sign of the intermediate result, not the final magnitude. For the angle calculations, however, the labelling (which vertex is A, B, or C) determines which angle is reported for which vertex.
How do I find the perimeter from coordinates?
Compute the length of each side with the distance formula: length = sqrt((x2 - x1)^2 + (y2 - y1)^2). The perimeter is the sum of all three side lengths. This calculator shows all three side lengths and their sum.
What does it mean if the area is zero?
An area of zero means the three points are collinear - they all lie on a single straight line and do not form a triangle. Check whether one point is a duplicate of another, or whether the three points happen to be on the same line (for example, (0,0), (1,1), (2,2)).
Can I use negative coordinates?
Yes. The shoelace formula and the distance formula both work with any real-valued coordinates, including negative values. The area is always non-negative because of the absolute value in the formula.
What is the centroid?
The centroid is the geometric center of a triangle, computed as the average of the three vertex coordinates: G = ((x1+x2+x3)/3, (y1+y2+y3)/3). It is the point where the three medians (lines from each vertex to the midpoint of the opposite side) intersect. For a uniform triangular plate, it is also the center of mass.
How is the shoelace formula related to the cross product?
In 2-D, the magnitude of the cross product of two vectors equals twice the area of the triangle they form. The shoelace expression x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) is equivalent to the z-component of the cross product of vectors AB and AC, scaled appropriately. Both approaches give the same answer.