To calculate the volumes of the triangular prism and triangular pyramid, we first need to know the area of the base triangle.
Let’s denote:
- The area of the base triangle as \( A \) (in square centimeters).
- The height of both the prism and pyramid as \( h = 9 \) cm.
Volume of the Triangular Prism
The volume \( V \) of a prism is given by: \[ V = A \times h \]
Volume of the Triangular Pyramid
The volume \( V \) of a pyramid is given by: \[ V = \frac{1}{3} A \times h \]
Let's Calculate Each Volume
-
Triangular Prism: \[ V_{prism} = A \times h = A \times 9 \]
-
Triangular Pyramid: \[ V_{pyramid} = \frac{1}{3} A \times h = \frac{1}{3} A \times 9 = 3 A \]
Finding the Relationship between Volumes
The volume of the prism is three times the volume of the pyramid (since \( V_{prism} = 3 \times V_{pyramid} \)): \[ V_{prism} = 3 V_{pyramid} \]
Answers Based on Choices
From the responses:
- If \( V_{prism} = 36 \) cm³, then \( V_{pyramid} = 12 \) cm³ (not listed).
- If \( V_{prism} = 108 \) cm³, then \( V_{pyramid} = 36 \) cm³ (not listed).
- If \( V_{prism} = 72 \) cm³, then \( V_{pyramid} = 24 \) cm³ (not listed).
- If \( V_{prism} = 216 \) cm³, then \( V_{pyramid} = 72 \) cm³ (not listed).
The correct answer would follow the relationship derived above. Therefore:
- The options given do not directly reflect values for volumes that adhere to \( V_{prism} = 3 \times V_{pyramid} \).
If we assumed \( A = 12 \):
- Prism: \( 12 \times 9 = 108 \) cm³
- Pyramid: \( \frac{1}{3} \times 12 \times 9 = 36 \) cm³
Thus, the correct option based on our derivation should be: prism: 108 cm³; pyramid: 36 cm³.