The two formulas for the volume of a cylinder, \( V = \pi r^2 \cdot h \) and \( V = B \cdot h \), are related through the concept of the base area of the cylinder.
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Base Area (B): In the second formula, \( V = B \cdot h \), \( B \) represents the area of the base of the cylinder. For a cylinder, which has a circular base, the area \( B \) can be calculated using the formula for the area of a circle: \[ B = \pi r^2 \] Here, \( r \) is the radius of the circular base.
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Substitution: By substituting the expression for the area of the base \( B \) into the volume formula, we get: \[ V = B \cdot h = (\pi r^2) \cdot h = \pi r^2 \cdot h \]
Thus, both formulas ultimately describe the same quantity (the volume of the cylinder) but from different perspectives: one emphasizes the base area and height, while the other explicitly uses the formula for the area of the circular base. Therefore, they are essentially two ways of expressing the same mathematical relationship.