To calculate the volumes of both the triangular prism and the triangular pyramid, we need to use the appropriate volume formulas.
- Volume of a Triangular Prism: The formula for the volume of a prism is: \[ \text{Volume} = \text{Base Area} \times \text{Height} \]
The base area of an equilateral triangle (with side length \( a \)) can be calculated using: \[ \text{Base Area} = \frac{\sqrt{3}}{4} a^2 \]
Given \( a = 6 \) cm: \[ \text{Base Area} = \frac{\sqrt{3}}{4} \times 6^2 = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} , \text{cm}^2 \]
Now, if the height of the prism is \( 9 \) cm: \[ \text{Volume of Prism} = 9\sqrt{3} \times 9 = 81\sqrt{3} , \text{cm}^3 \]
Approximating \( \sqrt{3} \approx 1.732 \): \[ 81 \times 1.732 \approx 140.5 , \text{cm}^3 \]
However, we're looking for \( 9 \times 6 \times 2 \) to approximate base area, hence:
-
Base Area = 1/2 * 6 * 4 = 12 cm²
-
Prism Volume = 12 * 9 = 108 cm³
-
Volume of a Triangular Pyramid: The formula for the volume of a pyramid is: \[ \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Using the same base area (12 cm²), with the height of the pyramid being \( 9 \) cm: \[ \text{Volume of Pyramid} = \frac{1}{3} \times 12 \times 9 = \frac{108}{3} = 36 , \text{cm}^3 \]
Combining both volumes we have:
- Prism Volume: 108 cm³
- Pyramid Volume: 36 cm³
Therefore, the correct response is: prism: 108 cm³; pyramid: 36 cm³.