The center of gravity of an irregular shape can be measured, for example by hanging the object from two different points (points of the object), then watching where the lines (from the point where it is hung up downwards) intersect.
If you know details about the shape, the center of gravity can also be calculated by integration. Basically this means dividing (through calculation) the shape into small pieces, and adding up the results.
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Each body has its own centre of gravity. The centre of gravity of two regular shapes - an equilateral triangle and a square will be different so why should the cog of a regular and an irregular shape not be different?
Irregular shapes are all around. Most shapes are irregular.
Irregular 3-dimensional shapes.
Bodies that are small and light (examples: moons of Mars) have low gravity and tend to have irregular shapes. Above a certain size, however, gravity is strong enough to overcome the strength of rock, forcing the body into a spherical geometry that minimizes surface area-to-volume ratio.
Regular shapes are both equilateral and equiangular. Irregular shapes may or may not be equilateral and equiangular.