To informally argue for the volume of a pyramid based on the volume of a cube, let's start by understanding the relationship between the two shapes.
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Volume of the Cube: We are given that the volume of the cube is 12 cubic meters. The volume of a cube is calculated using the formula: \[ V = \text{side}^3 \] So, if we denote the length of each side of the cube as \( s \), we have: \[ s^3 = 12 , \text{m}^3 \]
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Pyramid Dimensions: Now, consider the square-based pyramid that has the same base area and height as the cube. The base area of the pyramid is the same as the area of the square face of the cube, which can be expressed as: \[ \text{Base Area of Pyramid} = s^2 \]
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Height of the Pyramid: The height of the pyramid is the same as the side length of the cube, which is \( s \).
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Volume of the Pyramid: The formula for the volume of a pyramid is given by: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \] Substituting the known values into this formula, we get: \[ V = \frac{1}{3} \times s^2 \times s = \frac{1}{3} s^3 \]
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Relate to Cube Volume: Since we established earlier that \( s^3 = 12 , \text{m}^3 \), we can substitute this into the volume calculation for the pyramid: \[ V = \frac{1}{3} \times 12 = 4 , \text{m}^3 \]
Thus, the volume of the pyramid is \( 4 , \text{m}^3 \).
So the answer is 4 m³ or 4 m cubed.