To find the volume of the triangular prism, we can use the formula for the volume of a prism:
\[ V = \text{Base Area} \times \text{Height} \]
Given that the triangular face (base area) is 21 cm² and the height (the length connecting the two triangular faces) is 7 cm, we can calculate the volume:
\[ V = 21 , \text{cm}^2 \times 7 , \text{cm} = 147 , \text{cm}^3 \]
So, the volume of the prism is 147 cm³.
Now, let's calculate the volume of the triangular pyramid using the formula for the volume of a pyramid:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Using the same base area of 21 cm² and height of 7 cm:
\[ V = \frac{1}{3} \times 21 , \text{cm}^2 \times 7 , \text{cm} = \frac{1}{3} \times 147 , \text{cm}^3 = 49 , \text{cm}^3 \]
Thus, the volume of the pyramid is 49 cm³.
Now to compare the two volumes:
The volume of the prism and the volume of the pyramid are different because the cross sections of the pyramid are smaller at the top and tapering down to the apex, while the prism retains the same cross-sectional area throughout its height.
In conclusion:
- Volume of the prism: 147 cm³
- Volume of the pyramid: 49 cm³
The volumes differ, with the prism having three times the volume of the pyramid due to its constant cross-sectional area compared to the tapering cross section of the pyramid.