Quick answer
The orthocenter is the point where the three altitudes of a triangle meet. For a triangle with vertices (0, 0), (4, 0) and (0, 3), it lies at (0, 0), the right-angle vertex.
Enter the three vertices. The calculator returns the orthocenter.
Related tools: Area Calculator: Area of common two-dimensional shapes. Right Triangle Calculator: Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter. Pythagorean Theorem Calculator: Find the hypotenuse or a missing leg of a right triangle.
How to use the Orthocenter Calculator
- Enter x₁.
- Enter y₁.
- Enter x₂.
- Enter y₂.
- Enter x₃.
- Enter y₃.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| x₁ | A number | — |
| y₁ | A number | — |
| x₂ | A number | — |
| y₂ | A number | — |
| x₃ | A number | — |
| y₃ | A number | — |
Formula and method
Formula
Intersection of two altitudes
IEEE-754 double precision; shown to at most 6 decimal places.
Solves the two altitude equations as a 2×2 linear system. Collinear vertices are rejected.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: x₁ = 0; y₁ = 0; x₂ = 4; y₂ = 0; x₃ = 0; y₃ = 3.
| Result | Value |
|---|---|
| Orthocenter | (0, 0) |
Example 2
Inputs: x₁ = 0; y₁ = 0; x₂ = 4; y₂ = 0; x₃ = 1; y₃ = 3.
| Result | Value |
|---|---|
| Orthocenter | (1, 1) |
The first line is the headline result (orthocenter). IEEE-754 double precision; shown to at most 6 decimal places.
Where it lies
For an acute triangle the orthocenter is inside; for an obtuse triangle it is outside; for a right triangle it is at the right-angle vertex, as here. The altitudes from the two legs run along the legs and meet at the corner.
How it is found
The calculator intersects two altitudes (the lines through a vertex perpendicular to the opposite side). The triangle must be non-degenerate. Together with the centroid and circumcenter it lies on the Euler line.
How the result changes with x₂
Holding every other input at the first worked example, this table changes only x₂. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| x₂ | Orthocenter |
|---|---|
| 2 | (0, 0) |
| 4 | (0, 0) |
| 8 | (0, 0) |
| 16 | (0, 0) |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- x₃: entering "2" shows "The three points are collinear." beside that field and no result.
Choosing between the Orthocenter Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Area Calculator](/math/area-calculator/) | shape and a | Area |
| [Right Triangle Calculator](/math/right-triangle-calculator/) | leg a and leg b | Hypotenuse c |
| [Pythagorean Theorem Calculator](/math/pythagorean-theorem-calculator/) | find, leg a and other side | Hypotenuse c |
| [Triangle Calculator](/math/triangle-calculator/) | side a, side b and side c | Area |
| [Circle Calculator](/math/circle-calculator/) | i know the and value | Area |
Background: geometry measurements
Geometry results carry units that depend on the quantity: a perimeter or length is in linear units such as cm, an area or surface area is in square units (cm²) and a volume is in cubic units (cm³). Use one length unit for every input, because the calculator multiplies the numbers as given.
Flat (2D) shapes have an area and a perimeter; solids (3D) have a surface area and a volume. Two familiar checks: a circle of radius r has area πr² and circumference 2πr, and a sphere has volume (4/3)πr³ and surface area 4πr².
Assumptions and limitations
- Non-degenerate triangle
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for x₃ to confirm this.
Frequently asked questions
Is the orthocenter always inside?
No, only for acute triangles.
What is an altitude?
A perpendicular from a vertex to the opposite side.
Where is it for an equilateral triangle?
At the centre, same as the centroid.
What does the Orthocenter Calculator calculate?
Orthocenter of a triangle from its vertices. It returns Orthocenter.
What formula does the Orthocenter Calculator use?
Intersection of two altitudes. Solves the two altitude equations as a 2×2 linear system.
What do I need to enter in the orthocenter Calculator?
You enter x₁, y₁, x₂, y₂, x₃ and y₃.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Non-degenerate triangle.
What is an example of a orthocenter result?
With x₁ 0, y₁ 0, x₂ 4, y₂ 0 and x₃ 0, the orthocenter is (0, 0).
Related calculators
- Area Calculator — Area of common two-dimensional shapes.
- Right Triangle Calculator — Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
- Pythagorean Theorem Calculator — Find the hypotenuse or a missing leg of a right triangle.
- Triangle Calculator — Area, perimeter, angles and type of a triangle from its three sides.
- Circle Calculator — Radius, diameter, circumference and area of a circle from any one of them.
- Circumference Calculator — Circumference of a circle.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.