The two formulas for the volume of a cylinder are indeed related, and understanding their connection can clarify how the volume is derived.
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Formula 1: \( V = \pi r^2 h \)
This formula calculates the volume \( V \) of a cylinder using the radius \( r \) and height \( h \). Here, \( \pi r^2 \) represents the area of the circular base of the cylinder. Multiplying this area by the height \( h \) gives the total volume of the cylinder. -
Formula 2: \( V = B \times h \)
This formula defines the volume \( V \) in terms of the base area \( B \) multiplied by the height \( h \). In the case of a cylinder, the base \( B \) is the area of the circular cross-section, which is \( \pi r^2 \).
Relationship Between the Formulas
The volume formulas are essentially two perspectives on the same concept.
- In the first formula, you calculate the area of the base independently and then incorporate it into the overall volume calculation.
- In the second formula, you emphasize that volume can be found by multiplying any base area by the height, where the base area \( B \) happens to be \( \pi r^2 \) for a cylinder.
Thus, if you substitute \( B \) from the second formula into the first formula, you get:
\[ V = B \times h = (\pi r^2) \times h \]
This demonstrates that both formulas will yield the same volume for a cylinder, reinforcing the idea that the volume is determined by the area of the base and the height.
Diagram
If I were to use a sketch tool, I would visualize a cylinder showing:
- The circular base with radius \( r \), labeling the area as \( B = \pi r^2 \),
- A vertical height \( h \),
- The overall shape depicting that multiplying the base area by height forms the volume of the cylinder.
This visual representation would help clarify how the base area and height work together to define the volume.
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