Triangle Orthocenter Calculator

Enter the three vertex coordinates of a triangle on the Cartesian plane and get its orthocenter, altitude feet and lengths, side lengths, area, centroid, and its side/angle classification in one pass.

What is an orthocenter?

The orthocenter is the point where a triangle's three altitudes intersect. An altitude is the perpendicular line dropped from a vertex to the line containing the opposite side. Unlike the centroid, the orthocenter isn't always inside the triangle:

• Acute triangle — orthocenter is inside.
• Right triangle — orthocenter sits exactly at the right-angle vertex.
• Obtuse triangle — orthocenter is outside.

Formulas

For a triangle with vertices A(x₁,y₁), B(x₂,y₂), C(x₃,y₃):

Foot of altitude from A = orthogonal projection of A onto line BC
                          (and likewise for the feet from B and C)
Orthocenter             = intersection of line(A, footA) and line(B, footB)
Side a (BC)             = distance from B to C     (opposite vertex A)
Area                    = |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| / 2      (shoelace formula)
Altitude from A         = 2 × Area / side a

(and likewise for the sides/altitudes from B and C, each relative to its opposite side)

How to use it

1. Enter the x and y coordinates for each of the three vertices, A, B, and C.
2. Choose how many decimal places to display.
3. Results update instantly as you type.

FAQ

Why isn't the orthocenter always inside the triangle?

Because the altitude lines are extended through the whole line containing each side, not just the segment itself. In an obtuse triangle, two of those extended altitudes only cross each other outside the triangle's boundary.

What if my three points are on a straight line?

Three collinear points don't form a triangle — there's no enclosed area, so no altitudes or orthocenter to compute. The calculator flags this rather than returning a misleading result.

Does vertex order (A, B, C) matter?

No — the orthocenter, area, and classification come out identical no matter which order you enter the vertices in. Only the side/altitude labels (which one is "opposite A" vs "opposite B") shift accordingly.

I have side lengths and angles, not coordinates — can I still use this?

Use the Triangle Calculator instead — it solves a triangle from any three known sides/angles (SSS, SAS, ASA, AAS, or SSA) without needing coordinates. For the centroid instead of the orthocenter, see the Triangle Centroid Calculator.

Privacy

Everything runs locally in your browser — nothing you enter is uploaded or stored anywhere.