Question

8.8.PS-13
Find the surface area of the
regular hexagonal prism.
3.5 cm
The surface area is cm Superscript 2
4 cm
11 cm
...

Answer 1

To find the surface area of a regular hexagonal prism, we need to calculate the areas of its individual faces and then add them up.

A regular hexagonal prism has two hexagonal bases and six rectangular faces connecting the bases. The formula for the surface area of a regular hexagonal prism is:

Surface Area = 2 × Base Area + Lateral Area

1. Base Area:

The base of a regular hexagonal prism is a regular hexagon, and the formula for the area of a regular hexagon is:

Base Area = (3 × √3 × s²) / 2

Where "s" is the length of each side of the hexagon.

2. Lateral Area:

The lateral area of a regular hexagonal prism is the sum of the areas of all six rectangular faces. Since the opposite faces are congruent, you only need to calculate the area of one rectangle and multiply it by 6.

Lateral Area = Perimeter of Base × Height

The perimeter of the base can be calculated by multiplying the length of one side by 6, as a regular hexagon has six equal sides.

Now, let's calculate the surface area for the given measurements:

1. For a regular hexagonal prism with a side length of 3.5 cm:

Base Area = (3 × √3 × 3.5²) / 2

= (3 × √3 × 12.25) / 2

≈ 63.64 cm²

Lateral Area = Perimeter of Base × Height

= (6 × 3.5) × 11

= 231 cm²

Surface Area = 2 × Base Area + Lateral Area

= 2 × 63.64 + 231

≈ 358.28 cm²

Therefore, the surface area of the regular hexagonal prism with a side length of 3.5 cm is approximately 358.28 cm².

2. For a regular hexagonal prism with a side length of 4 cm:

Base Area = (3 × √3 × 4²) / 2

= (3 × √3 × 16) / 2

= 69.28 cm²

Lateral Area = Perimeter of Base × Height

= (6 × 4) × 11

= 264 cm²

Surface Area = 2 × Base Area + Lateral Area

= 2 × 69.28 + 264

= 402.56 cm²

Therefore, the surface area of the regular hexagonal prism with a side length of 4 cm is 402.56 cm².

In both cases, the surface area is expressed as cm².

Answer 2

Answer:

348 cm²

Step-by-step explanation:

In this problem, we are asked to find the surface area of the regular hexagonal prism. We can use the formula:

$SA = 2A + (P \cdot h)$

where $A$ is the area of the prism's hexagonal faces, $P$ is the perimeter of the hexagonal faces, and $h$ is the prism's height.

We can solve for the area of the hexagonal faces by splitting it into 6 triangles and multiplying the area of each triangle by 6.

$A = 6\left(\dfrac{1}{2} bh\right)$

where $b$ is the triangle's base and $h$ is the triangle's height.

$A = 6\left(\dfrac{1}{2}\cdot 4 \cdot 3.5\right)$

$A = 6\left(2 \cdot 3.5\right)$

$A = 12 \cdot 3.5$

$A = 42\text{ cm}^2$

We can solve for the perimeter by multiplying one side length by 6.

$P = 6(4) = 24\text{ cm}$

We are given that the height of the prism is 11 cm.

$h = 11\text{ cm}$

Finally, we can solve for the surface area of the prism by plugging these values into the above formula.

$SA = 2A + (P \cdot h)$

$SA = 2(42) + (24 \cdot 11)$

$SA = 84 + 264$

$\boxed{SA = 348 \text{ cm}^2}$