It seems that many badges look like shields because a shield is symbolic of protection. However, many badges are in the shape of a star with varying numbers of points. So where did the use of a star-shaped badge come from? Is the star symbolic in some way?
Shire-Reeves in England wore metallic star shaped badges that could be highly polished. It allowed them to easily identify each other. Even at night due to the moonlight reflecting off the badge because they were so highly polished.
When the US began establishing organized police departments, they pulled from the tradition established by the Shire-Reeve. Not all departments in the US use the star shaped. Some departments (LAPD and NYPD, for example) use a shield.
Badges are unique to each department and every aspect of the design has a specific meaning. Even how many points of the star has a meaning. For example, the California Highway Patrol's badge has 7 points on their badge with stand for; Character, Integrity, Judgement, Loyalty, Courtesy, Honor, and Knowledge.