Wikipedia:Reference desk/Archives/Mathematics/2023 February 10

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February 10

Average height of a triangular segment

If I have a triangle in 3d space defined by three points (x0, y0, z0), (x1, y1, z1), (x2, y2, z2) is the average height (defined as the distance from the XY plane) of all the points in the triangle simply the average of the Z values? It seems like it should be, but I just can't seem to justify that to myself. 2A01:E34:EF5E:4640:6279:1CCE:459D:DEE0 (talk) 09:13, 10 February 2023 (UTC)

I think the answer is NO. The "average height" will be the z-coordinate of the centroid of the triangle; and I don't think that will be the average of the three Z-coordinates. Dolphin (t) 11:31, 10 February 2023 (UTC)

But the centroid of a triangle is the mean of its vertices - so yes, the mean z is the mean of z₀, z₁ and z₂. catslash (talk) 16:11, 10 February 2023 (UTC)

The formula for the centroid of a general polygon is more complicated, not just the average of the vertices. They happen to work out the same for a triangle, but the reason is not immediately obvious. --RDBury (talk) 17:28, 10 February 2023 (UTC)