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You can get through many aspects of geometry without pre-algebra or algebra. However, when it comes to the measurement in geometry, you need algebra for that.
Algebra must be learned before calculus. Concepts that are learned in algebra are used in calculus, to the extent that a student cannot succeed in calculus unless he knows algebra so well that he does it without thinking.Algebra is the study of constants and variables; that is, it is the study of numbers without knowing specifically what those numbers are.Calculus is the study of rates of change, and is done almost entirely abstractly (without using specific numbers), so it cannot be done without the use of constants and variables (algebra).
Pre-algebra is essentially the basics of algebra. Algebra can be tough for many, so as a way to break the ice, you (ussually) take a pre-algebra course before algebra itself. Algebra is a very large subject, spanning multiple years of courses in most (if not all) schools. After pre-algebra, you have to work with many algebraic equations in the later courses, which cannot be understood without learning the basics that are taught in pre-algebra.
We use it all the time without knowing it. When we go to the store or do finances, it's all algebra. Engineering and many scientific fields also use algebra.