General formula for centre of mass of polygons

[Happiness]

The position vector of the center of mass of a triangle is $\frac{1}{3}(\mathbf{a}+\mathbf{b}+\mathbf{c})$.

Is the position vector of the center of mass of a planar four-sided figure ABCD $\frac{1}{4}(\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d})$? Does this generalise to n-sided figure?

How about a 3D figure ABCD in the form of an arbitrary triangular pyramid (whose faces need not be equilateral)? Is the center of mass also $\frac{1}{4}(\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d})$? Does this generalise to n-dimensonal figure?

 

[mfb]

Yes to all.