Examples of calculating the volume of hexagonal prisms with given dimensions and formulas.

Here is the step-by-step solution to fill in the blanks for both examples.

Example 1

1. Find the Area of the Base ($B$):
The base is a regular octagon. The formula for the area of a regular polygon when you know the side length and the apothem (the distance from the center to the middle of a side) is:
$$ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} $$
Or simply: $B = \frac{1}{2} P a$

* Side length ($s$): $8 \text{ cm}$
* Number of sides: $8$
* Perimeter ($P$): $8 \text{ sides} \times 8 \text{ cm/side} = 64 \text{ cm}$
* Apothem ($a$): $9.7 \text{ cm}$

Now, calculate $B$:
$$ B = \frac{1}{2} \times 64 \times 9.7 $$
$$ B = 32 \times 9.7 $$
$$ B = 310.4 \text{ cm}^2 $$

2. Identify the Height ($h$):
Looking at the diagram, the vertical height of the prism is labeled as $19 \text{ cm}$.

3. Calculate the Volume ($V$):
The formula for the volume of a prism is $V = B \cdot h$.
$$ V = 310.4 \times 19 $$
$$ V = 5897.6 \text{ cm}^3 $$

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Example 2

1. Find the Area of the Base ($B$):
The base is a regular hexagon. We use the same area formula: $B = \frac{1}{2} P a$.

* Side length ($s$): The diagram shows the full width across opposite vertices is $8.1 \text{ m}$. For a regular hexagon, the distance across opposite vertices is equal to $2 \times \text{side length}$. Therefore, the side length is half of that: $8.1 / 2 = 4.05 \text{ m}$.
*(Note: In many textbook problems like this, if a single dimension like 8.1 is given pointing to the shape's extent, it often represents the "diameter" or width. Let's assume the side length is derived from this or that 8.1 is the specific dimension needed for the perimeter calculation directly if interpreted differently. However, looking closely at the arrow for 8.1m, it spans the entire width of the hexagon. So Side = 4.05m is correct.)*

*Correction/Alternative Interpretation:* Sometimes diagrams label the side length directly. If 8.1m were the side length, the perimeter would be huge. Given the visual proportion where the height is 7m and width is 8.1m, it is highly likely 8.1m is the total width (point-to-point).
Let's re-read the diagram carefully. The arrow for 8.1m goes from one flat side to the opposite flat side? No, it goes vertex to vertex.
Wait, let's look at Example 1 again. It gave side=8 and apothem=9.7.
In Example 2, we have an apothem labeled $7 \text{ m}$. We need the Perimeter.
The label $8.1 \text{ m}$ has arrows pointing to the left-most and right-most points of the hexagon. This is the distance between opposite vertices.
In a regular hexagon, Distance (vertex-to-vertex) = $2 \times \text{Side Length}$.
So, $\text{Side Length} = 8.1 / 2 = 4.05 \text{ m}$.
$\text{Perimeter} (P) = 6 \times 4.05 = 24.3 \text{ m}$.

Now calculate Area ($B$):
$$ B = \frac{1}{2} \times P \times a $$
$$ B = \frac{1}{2} \times 24.3 \times 7 $$
$$ B = 12.15 \times 7 $$
$$ B = 85.05 \text{ m}^2 $$

*(Self-Correction Check: What if 8.1 is the side length? If side=8.1, P=48.6. B = 0.5 * 48.6 * 7 = 170.1. Visually, the apothem (7) looks similar in length to the side. In a regular hexagon, apothem = side * $\sqrt{3}/2$ ≈ side * 0.866. If side was 8.1, apothem would be ~7.01. This matches the label "7 m" very closely. It is extremely likely that 8.1 m is the side length, not the total width, despite the slightly ambiguous arrow placement which might just be indicating the dimension of the face. Let's proceed with Side Length = 8.1 m because the math works out perfectly with standard hexagon ratios ($8.1 \times 0.866 \approx 7.01$).)*

Revised Calculation for Example 2 assuming Side = 8.1 m:
* Side ($s$): $8.1 \text{ m}$
* Perimeter ($P$): $6 \times 8.1 = 48.6 \text{ m}$
* Apothem ($a$): $7 \text{ m}$
* Base Area ($B$): $\frac{1}{2} \times 48.6 \times 7 = 24.3 \times 7 = 170.1 \text{ m}^2$

2. Identify the Height ($h$):
The length of the prism is labeled as $21 \text{ m}$.

3. Calculate the Volume ($V$):
$$ V = B \cdot h $$
$$ V = 170.1 \times 21 $$
$$ V = 3572.1 \text{ m}^3 $$

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Final Answer

EXAMPLE 1
* $B = \mathbf{\frac{1}{2}(64)(9.7)} = \mathbf{310.4}$
* $h = \mathbf{19}$
* So $V = B \cdot h = \mathbf{310.4 \cdot 19}$
* $V = \mathbf{5897.6} \text{ cm}^3$

EXAMPLE 2
* $B = \mathbf{\frac{1}{2}(48.6)(7)} = \mathbf{170.1}$
* $h = \mathbf{21}$
* So $V = B \cdot h = \mathbf{170.1 \cdot 21}$
* $V = \mathbf{3572.1} \text{ m}^3$