How can you locate the centroid of a convex quadrilateral?

Answer 1

The analytical method is far from simple:

Divide the quadrilateral into two triangles, say ABC and BCD.

The centroid of triangle ABC is P = [(xa+xb+xc)/3 , (ya+yb+yc)/3] where (xa,ya) are the coordinates of A and so on.

Label the centroid of BCD as Q and find its coordinates in a similar way.

The next step is to find the areas of the two triangles.

Area (ABC) = g

Area(BCD) = h

Then the centroid of the whole quadrilateral is the weighted average of the coordinates of P and Q, with weights g and h.

Thus, if R is the centroid of the quadrilateral, then

xr = (g*xp + h*xq)/(g+h) and

yr = (g*yp + h*yq)/(g+h).

If you can cut out the quadrilateral the procedure is much simpler. Suspend the quadrilateral from one corner and, using a plumb line or equivalent, mark the vertical direction. Repeat from the adjacent vertex. The two lines will meet at the centroid.