The two formulas for the volume of a cylinder, \( V = \pi r^2 \cdot h \) and \( V = B \cdot h \), are related in the way they define volume based on different parameters.
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Definition of Base Area (B): In the formula \( V = B \cdot h \), \( B \) represents the area of the base of the cylinder. For a cylinder with a circular base, the area \( B \) can be calculated using the formula for the area of a circle, which is \( B = \pi r^2 \).
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Substitution: When you substitute the formula for the area of the circular base into the volume formula, you get: \[ V = B \cdot h = \pi r^2 \cdot h. \] This shows that the first formula is a specific case of the second formula, where the base is a circle.
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Generalization: The formula \( V = B \cdot h \) is more general, as it can be applied to any shape of the base (not just circles), by simply calculating the corresponding area of that particular shape and using it in the formula.
In summary, both formulas express the same concept of volume: the volume of a cylinder is the area of its base multiplied by its height. The formula \( V = \pi r^2 \cdot h \) is specifically for a cylinder with a circular base, while \( V = B \cdot h \) is a broader formula applicable to any cylindrical shape.
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