## Most recent change of Centroid

Edit made on October 13, 2015 by GarethMcCaughan at 00:44:10

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~~WW~~ WM

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[[[>50 IMG:Centroid.png ]]]

The centroid of a triangle is located at the intersection of the medians of the triangle. Depending on context, this point is also ~~know~~ known as the centre of mass or centre of gravity of the triangle. If you make a model of the triangle out of some flat, rigid material, the centroid is the point at which you can balance it on a point: for any line through the centroid, the total torque produced about that line by gravity is zero.

Taking that last sentence as the definition of "centroid" for any two-dimensional shape, every such shape has a centroid. (I am handwaving a bit about the definition of "shape" here.) Here is a very rough sketch of a proof. For each point in the shape, consider all lines through that point and look at the biggest net torque produced by the action of gravity on the shape. Find a point that minimizes this number. The number there must be zero, because otherwise you can decrease it by moving perpendicular to a line through that point that gives the maximum torque.

The centroid is the "mean position" of all points in the shape, in the following sense: it equals the integral of (position) d(area) over the whole shape, divided by the integral of 1 d(area). (Imagine dividing the shape into many tiny regions of equal area and averaging their locations. Or dividing it into many tiny regions of not-necessarily-equal area, and averaging their locations weighted by their areas.)

Shapes in more than two dimensions also have centroids.