Triangle Vertices Calculator

Enter 3 vertex coordinates to compute side lengths, angles, area, perimeter, centroid, circumcenter, incenter, and orthocenter.

Enter Vertex Coordinates

Results

Area
--
square units
Side Lengths
Angles
Properties
Special Points

Step-by-Step Solution

Area = 0.5 |x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B)|

Triangle from Vertex Coordinates

A triangle defined by three vertices in a coordinate plane can be fully analyzed using coordinate geometry. Given points A(x1, y1), B(x2, y2), and C(x3, y3), you can determine every property of the triangle including its side lengths, interior angles, area, and several important centers.

Key Formulas

Distance Formula

Find the length of each side using the distance between two points.

d = sqrt((x2-x1)^2 + (y2-y1)^2)

Shoelace Formula (Area)

Compute the area directly from vertex coordinates.

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

Law of Cosines (Angles)

Find each angle from the three known side lengths.

cos(A) = (b^2+c^2-a^2) / (2bc)

Centroid

The intersection of the three medians, always inside the triangle.

G = ((x1+x2+x3)/3, (y1+y2+y3)/3)

Circumcenter

Equidistant from all three vertices; center of the circumscribed circle.

Intersection of perpendicular bisectors

Orthocenter

The intersection of the three altitudes of the triangle.

Intersection of altitude lines

Triangle Centers Explained

Every triangle has four classical centers: the centroid (center of mass), the circumcenter (center of the circumscribed circle), the incenter (center of the inscribed circle), and the orthocenter (intersection of altitudes). The centroid always lies inside the triangle, as does the incenter. The circumcenter and orthocenter may lie outside for obtuse triangles.

Practical Applications

  • Surveying and land measurement using GPS coordinates
  • Computer graphics for mesh and polygon calculations
  • Navigation and triangulation methods
  • Structural engineering and truss analysis
  • Game development collision detection