Quick answer
The centroid of a triangle is the average of its vertices: G = ((x₁ + x₂ + x₃) ÷ 3, (y₁ + y₂ + y₃) ÷ 3). Vertices (0,0), (6,0) and (0,9) give a centroid of (2, 3) and an area of 27.
Enter the three vertices. The calculator returns the centroid and the area.
Related tools: Triangle Calculator: Area, perimeter, angles and type of a triangle from its three sides. 45-45-90 Triangle Calculator: Sides of a 45°-45°-90° triangle from one leg. 30-60-90 Triangle Calculator: Sides of a 30°-60°-90° triangle from the short leg.
How to use the Triangle Centroid 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
G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
IEEE-754 double precision; shown to at most 6 decimal places.
Average of the vertex coordinates.
Worked examples
This example is verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: x₁ = 0; y₁ = 0; x₂ = 6; y₂ = 0; x₃ = 0; y₃ = 9.
| Result | Value |
|---|---|
| Centroid | (2, 3) |
| Area | 27 |
The first line is the headline result (centroid). Below it the triangle Centroid Calculator lists area, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
The balance point
The centroid is where a cardboard triangle balances on a pin. It's also where the three medians meet, and it divides each median in a 2:1 ratio measured from the vertex.
Collinear points
If the three points lie on a line there is no triangle, and the area is reported as 0. The area comes from ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)| = 27 here.
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 | Centroid | Area |
|---|---|---|
| 3 | (1, 3) | 13.5 |
| 6 | (2, 3) | 27 |
| 12 | (4, 3) | 54 |
| 24 | (8, 3) | 108 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- y₃: entering "a" shows "y₃ must be a number." beside that field and no result.
Choosing between the Triangle Centroid 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 |
|---|---|---|
| [Triangle Calculator](/math/triangle-calculator/) | side a, side b and side c | Area |
| [45-45-90 Triangle Calculator](/math/45-45-90-triangle-calculator/) | leg length | Leg |
| [30-60-90 Triangle Calculator](/math/30-60-90-triangle-calculator/) | short leg length | Short leg |
| [Right Triangle Calculator](/math/right-triangle-calculator/) | leg a and leg b | Hypotenuse c |
| [Triangle Area Calculator](/math/triangle-area-calculator/) | base and height | 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
- Collinear points have no triangle (area 0 is reported)
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 y₃ to confirm this.
Frequently asked questions
Is the centroid always inside?
Yes. For a triangle the centroid always lies inside, two-thirds of the way along each median from a vertex.
What is a median?
A line from a vertex to the midpoint of the opposite side.
Is it the same as the circumcenter?
No, except for equilateral triangles.
What does the Triangle Centroid Calculator calculate?
Centroid of a triangle from its vertices. It returns Centroid and Area.
What formula does the Triangle Centroid Calculator use?
G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3). Average of the vertex coordinates.
What do I need to enter in the triangle Centroid 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. Collinear points have no triangle (area 0 is reported).
What is an example of a triangle Centroid result?
With x₁ 0, y₁ 0, x₂ 6, y₂ 0 and x₃ 0, the centroid is (2, 3).
Related calculators
- Triangle Calculator — Area, perimeter, angles and type of a triangle from its three sides.
- 45-45-90 Triangle Calculator — Sides of a 45°-45°-90° triangle from one leg.
- 30-60-90 Triangle Calculator — Sides of a 30°-60°-90° triangle from the short leg.
- Right Triangle Calculator — Solve a right triangle from its two legs: hypotenuse, angles, area and perimeter.
- Triangle Area Calculator — Area of a triangle from base and height.
- SAS Triangle Calculator — Solve a triangle from two sides and the included angle.
- SSS Triangle Calculator — Angles and area of a triangle from its three sides.
- Forty-Five Ninety Triangle Calculator — Sides of a 45-45-90 triangle from one leg.
- Isosceles Triangle Calculator — Height, area and angles of an isosceles triangle from its base and equal sides.
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.