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.
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Circles, some triangles, and irregular polygons
They need to convert rotational motion to motion of some other form - usually linear motion. That can only be accomplished through the use of specific irregular shapes.
Any regular polygon. And some irregular polygons which will have some adjacent sides equal - for example, a kite or an arrowhead (chevron).
There are a lot more than 4 types. Some simple examples: regular polyhedra, other polyhedra, star shapes, ellipoids (including spheres), toroids (like cycle inner tubes), shapes with two "holes" in them, totally irregular blobby shapes.
You get the area by using formulas. There is usually a specific formula to find the area of each shape. Some irregular shaps may not have a formula.