The point where the three medians of the triangle intersect.

The 'center of gravity' of the triangle

One of a triangle's points of concurrency.

The 'center of gravity' of the triangle

One of a triangle's points of concurrency.

Try this Drag the orange dots at A,B or C and note where the centroid is for various triangle shapes.

Refer to the figure above. Imagine you have a triangular metal plate, and try and balance it on a point - say a pencil tip. Once you have found the point at which it will balance, that is the centroid.

The centroid of a triangle is the point through which all the mass of a triangular plate seems to act. Also known as its 'center of gravity' , 'center of mass' , or barycenter.

A **fascinating fact** is that the centroid is the point where the triangle's
medians
intersect. See
medians of a triangle for more information. In the diagram above, the medians of the triangle are shown as dotted blue lines.

- The centroid is always inside the triangle
- Each median divides the triangle into two smaller triangles of equal area.
- The centroid is exactly two-thirds the way along each median.

Put another way, the centroid divides each median into two segments whose lengths are in the ratio 2:1, with the longest one nearest the vertex. These lengths are shown on the one of the medians in the figure at the top of the page so you can verify this property for yourself.

If you know the coordinates of the triangle's vertices, you can calculate the coordinates of the centroid. See Coordinates of centroid.

Incenter |
Located at intersection of the
angle bisectors.
See |

Circumcenter |
Located at intersection of the perpendicular bisectors of the sides See |

Centroid |
Located at intersection of the medians |

Orthocenter |
Located at intersection of the altitudes |

For more, and an interactive demonstration see Euler line definition.

- Make any triangle from heavy cardboard. Make it about 12 - 24" wide.
- Mark a point half way along each side.
- Draw a line from each midpoint to the opposite corner. These are the medians of the triangle. They should meet at a point - the centroid.
- Make a small hole at the centroid and thread a knotted string through it.
- When held up and suspended by the string it should balance (tricky to get it exactly balanced, but you should get close).
- Explain why.

- Triangle definition
- Hypotenuse
- Triangle interior angles
- Triangle exterior angles
- Triangle exterior angle theorem
- Pythagorean Theorem
- Proof of the Pythagorean Theorem
- Pythagorean triples
- Triangle circumcircle
- Triangle incircle
- Triangle medians
- Triangle altitudes
- Midsegment of a triangle
- Triangle inequality
- Side / angle relationship

- Perimeter of a triangle
- Area of a triangle
- Heron's formula
- Area of an equilateral triangle
- Area by the "side angle side" method
- Area of a triangle with fixed perimeter

- Right triangle
- Isosceles triangle
- Scalene triangle
- Equilateral triangle
- Equiangular triangle
- Obtuse triangle
- Acute triangle
- 3-4-5 triangle
- 30-60-90 triangle
- 45-45-90 triangle

- Incenter of a triangle
- Circumcenter of a triangle
- Centroid of a triangle
- Orthocenter of a triangle
- Euler line

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