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Centroid of a Triangle
From Latin: centrum - "center", and Greek: -oid -"like"
The point where the three medians of the triangle intersect.
The 'center of gravity' of the triangle
Try this Drag the orange dots at A,B or C and note where the centroid is for various triangle shapes.
(If there is no image below, see support page.)

Refer to the figure on the right. 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.

Interesting facts

  • 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.
Summary of triangle centers
There are many types of triangle centers. Below are three of the most common.
Incenter Located at intersection of the angle bisectors.   See Triangle incenter definition
Circumcenter Located at intersection of the perpendicular bisectors of the sides   See Triangle circumcenter definition
Centroid Located at intersection of medians  
Equilateral triangle centers
In the case of an equilateral triangle, all three of the above centers occur at the same point.
Try this
  1. Make any triangle from heavy cardboard. Make it about 12 - 24" wide.
  2. Mark a point half way along each side.
  3. 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.
  4. Make a small hole at the centroid and thread a knotted string through it.
  5. When held up and suspended by the string it should balance (tricky to get it exactly balanced, but you should get close).
  6. Explain why.

Related triangle topics

General

Triangle types

Triangle centers

Congruence and Similarity

Triangle quizzes and exercises

(C) 2007 Copyright John Page