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I am teaching my daughter geometry at home, she is a 6th grader age 12. Her teacher and school textbooks sometimes take a shortcut and give formulas to her but no good explanations. Is there a good explanation I can give to my daughter as to why the volume of a cone or pyramid or a pile of sand, etc. is 1/3rd the volume of a base surface area time height. We understand the base x height part but why 1/3rd. Can I explain this to her in any simple way without going into the usual calculus explanation of cutting the shape in smaller shapes (e.g. small cylinders of increasingly smaller base and taking it to infinity). I just don't think telling a student to just memorize or say that fill up sand in the shape and watch that it takes 3 fill ups to fill up a solid cube with the sand from a cone or pyramid of the same base is good enough. That is what her text book does and the class teacher offers nothing more.
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