Asked by Rhyan
Find the volume of the prism, and then compare it to a pyramid.
First, find the volume of the prism.
A triangular prism. One triangular face is labeled as 21 square cm, and the segment connecting the two identical faces is labeled 7 cm.
The volume is
cm3.
Great job!
Here’s a pyramid that also has a height of 7 cm, and a base with an area of 21 cm2. What can you conclude about the volume of the prism and the volume of the pyramid?
A triangular prism. One triangular face is labeled as 21 square cm, and the segment connecting the two identical faces is labeled 7 cm.
A triangular pyramid. the base is labeled as 21 square cm, and the height is labeled 7 cm.
The volume of the prism and pyramid are
because the cross sections of the pyramid are
.
First, find the volume of the prism.
A triangular prism. One triangular face is labeled as 21 square cm, and the segment connecting the two identical faces is labeled 7 cm.
The volume is
cm3.
Great job!
Here’s a pyramid that also has a height of 7 cm, and a base with an area of 21 cm2. What can you conclude about the volume of the prism and the volume of the pyramid?
A triangular prism. One triangular face is labeled as 21 square cm, and the segment connecting the two identical faces is labeled 7 cm.
A triangular pyramid. the base is labeled as 21 square cm, and the height is labeled 7 cm.
The volume of the prism and pyramid are
because the cross sections of the pyramid are
.
Answers
Answered by
GPT-4o mini
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.
\[
V = \text{Base Area} \times \text{Height}
\]
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.