Centroid and Center of Gravity

The Centroid is the average position of all the points of an object.

Irregular blue shape with a red dot marking its centroid, the balancing point.

When we cut a plane shape from a piece of card it balances perfectly on its centroid.

The geographic center of the USA (near Lebanon, Kansas) was found this way in 1918.

Center of Gravity

The Center of Gravity is the same as the centroid when the density is the same throughout.

They are often marked by a cross or dot and sometimes the letters CG or just G

3D torus (donut shape) with a red dot at its geometric center, representing its centroid.
For a torus the centroid is at the very center
(even though there's no part of the torus there!)

3D cone with a red dot marking its centroid, located one-quarter of the way up from the base on the central axis.
For a right solid cone the centroid is on
the center line and ¼ of the way from the base

Side profile of a car, illustrating that its complex structure makes finding a precise centroid difficult.

Just like the torus and the cone, a car also has a center of gravity somewhere inside it (often around the middle of the car, around the floor).

It's center of gravity can be hard to figure out, as there's lots of empty space and materials of different density (such as the engine vs the seats).

When doing calculations we can often replace an object with its center of gravity.

Sequence of a falling hammer rotating, showing its center of gravity following a straight vertical path.

Example: You drop a hammer!

It may spin a little, but its center of gravity will fall straight down.

It also drops faster and faster due to gravity.

(The only complication is air resistance, which affect its motion more as it goes faster.)

A force that goes through the center of gravity won't cause any rotation. In fact you can balance an object by supporting it directly below its center of gravity.

A hammer balanced on a finger, with the finger placed directly below the hammer's center of gravity.

Centroid vs. Center of Gravity vs. Center of Mass

  • Centroid is the geometric center (strictly about shape)
  • Center of Mass is where the mass is balanced (strictly about matter)
  • Center of Gravity is where the weight is balanced (about gravity's pull)

If an object has the same density throughout, and gravity is uniform, they are all at the exact same point!

Centroid of a Triangle

For a triangle we can find the centroid very easily:

  1. Find the midpoint of each side of the triangle
  2. Draw a line from each midpoint to the corner (vertex) of the opposite side. These lines are called medians
  3. The three medians cross at the centroid of the triangle

Try moving points A, B or C:

images/triangle.js?mode=median
centroid

An Amazing Property

The centroid divides each median into a 2:1 ratio (from the vertex to the opposite side).

Draw one yourself and measure!

7480, 7484, 11959, 11961, 17567, 17566, 17570, 7485, 13478, 17564
Geometry Index