Find the centre of mass of rectangular, triangular and circular laminae

Centre of Mass of Rectangular Laminae

A lamina refers to a two-dimensional, thin plate with uniform distribution of mass across it.
The centre of mass of a rectangular lamina lies at its geometric centre because of its uniform density.
This point is found by drawing diagonals from the corners of the rectangle. The point of intersection of these diagonals represents the centre of mass.
Thus, the centre of mass of a rectangle with corners at coordinates (0,0), (a,0), (0,b) and (a,b) is at (a/2, b/2), exactly in the middle of the rectangle.

For a triangular lamina, the centre of mass, also known as the centroid, is located at the point of intersection of the medians of the triangle.
The medians of a triangle are the lines drawn from each vertex to the midpoint of the opposite side.
The coordinates of the centre of mass can be determined through finding the average of the x-coordinates and y-coordinates of the vertices of the triangle.
Specifically, if the three vertices of the triangle are given by coordinates (x1, y1), (x2, y2) and (x3, y3) then the position of the centre of mass (X, Y) is given by X = (x1 + x2 + x3) / 3 Y = (y1 + y2 + y3) / 3

For a circular lamina (such a thin disc), the centre of mass lies at the geometric centre of the circle due to its symmetry and uniform density.
This is true regardless of whether the circular lamina is full, or only a sector (portion) of a full circle.
For an arc or sector of a circle, it is best to consider it as a full disc but with a missing segment, so the centre of mass is computed by considering the mass distribution of the missing segment.
Thus the calculation normally involves a combination of both integration to find the centre of mass of the full disc, and subtraction to account for the removed segment.