Question
Question content area top
Part 1
To the nearest cubic centimeter, what is the volume of the regular hexagonal prism?
7 cm
14 cm
6.1 cm
Question content area bottom
Part 1
The volume of the regular hexagonal prism is about
enter your response here cm cubed .
(Do not round until the final answer. Then round to the nearest whole number as needed.)
The volume of the regular hexagonal prism is aboutcm cubed .(Do not round until the final answer. Then round to the nearest whole number as needed.)
Help me solve this
Part 1
To the nearest cubic centimeter, what is the volume of the regular hexagonal prism?
7 cm
14 cm
6.1 cm
Question content area bottom
Part 1
The volume of the regular hexagonal prism is about
enter your response here cm cubed .
(Do not round until the final answer. Then round to the nearest whole number as needed.)
The volume of the regular hexagonal prism is aboutcm cubed .(Do not round until the final answer. Then round to the nearest whole number as needed.)
Help me solve this
Answers
GPT-4o mini
To find the volume of a regular hexagonal prism, you can use the formula:
\[ V = B \cdot h \]
where:
- \( V \) is the volume of the prism,
- \( B \) is the area of the base (which is a regular hexagon),
- \( h \) is the height of the prism.
### Step 1: Calculate the Area of the Hexagonal Base
The area \( B \) of a regular hexagon can be calculated using the formula:
\[ B = \frac{3\sqrt{3}}{2} s^2 \]
where \( s \) is the length of a side of the hexagon.
Given:
- The side length \( s = 7 \, \text{cm} \)
Now, let's calculate the area of the hexagonal base:
\[
B = \frac{3\sqrt{3}}{2} (7^2) = \frac{3\sqrt{3}}{2} (49) = \frac{147\sqrt{3}}{2}
\]
### Step 2: Use the Height to Compute the Volume
Now, we have the height \( h = 14 \, \text{cm} \).
Now multiply the area of the base by the height:
\[
V = B \cdot h = \left(\frac{147\sqrt{3}}{2}\right) \cdot 14
\]
Calculating this gives:
\[
V = \frac{147\sqrt{3}}{2} \cdot 14 = 147 \cdot 7\sqrt{3} = 1029\sqrt{3}
\]
### Step 3: Calculate the Numerical Value
Using the approximate value of \(\sqrt{3} \approx 1.732\):
\[
V \approx 1029 \cdot 1.732 \approx 1788.5
\]
### Step 4: Round to the Nearest Whole Number
Thus, rounding \( 1788.5 \) to the nearest whole number gives:
\[
V \approx 1789 \, \text{cm}^3
\]
### Final Answer
So, the volume of the regular hexagonal prism is about **1789 cm³**.
\[ V = B \cdot h \]
where:
- \( V \) is the volume of the prism,
- \( B \) is the area of the base (which is a regular hexagon),
- \( h \) is the height of the prism.
### Step 1: Calculate the Area of the Hexagonal Base
The area \( B \) of a regular hexagon can be calculated using the formula:
\[ B = \frac{3\sqrt{3}}{2} s^2 \]
where \( s \) is the length of a side of the hexagon.
Given:
- The side length \( s = 7 \, \text{cm} \)
Now, let's calculate the area of the hexagonal base:
\[
B = \frac{3\sqrt{3}}{2} (7^2) = \frac{3\sqrt{3}}{2} (49) = \frac{147\sqrt{3}}{2}
\]
### Step 2: Use the Height to Compute the Volume
Now, we have the height \( h = 14 \, \text{cm} \).
Now multiply the area of the base by the height:
\[
V = B \cdot h = \left(\frac{147\sqrt{3}}{2}\right) \cdot 14
\]
Calculating this gives:
\[
V = \frac{147\sqrt{3}}{2} \cdot 14 = 147 \cdot 7\sqrt{3} = 1029\sqrt{3}
\]
### Step 3: Calculate the Numerical Value
Using the approximate value of \(\sqrt{3} \approx 1.732\):
\[
V \approx 1029 \cdot 1.732 \approx 1788.5
\]
### Step 4: Round to the Nearest Whole Number
Thus, rounding \( 1788.5 \) to the nearest whole number gives:
\[
V \approx 1789 \, \text{cm}^3
\]
### Final Answer
So, the volume of the regular hexagonal prism is about **1789 cm³**.