https://file.aiquickdraw.com/mathbot/file/1725918633534lt38hdd3.jpg

The image shows a hexagonal prism, with dimensions given as follows:

- The height of the prism is \( 10 \, \text{cm} \),
- The side length of the hexagonal base is \( 5 \, \text{cm} \),
- The width of the hexagon across the flats is \( 6 \, \text{cm} \).

We are asked to find the total surface area \( A_T \) of the prism.

### Surface Area of a Prism
The total surface area of a prism consists of two parts:
1. The lateral surface area, which is the area of the sides.
2. The area of the two hexagonal bases.

#### Step 1: Calculate the lateral surface area
The lateral surface area \( A_L \) of a prism is given by:
\[
A_L = \text{Perimeter of the base} \times \text{Height}
\]
For a regular hexagon, the perimeter \( P \) is:
\[
P = 6 \times \text{side length} = 6 \times 5 = 30 \, \text{cm}
\]
Thus, the lateral surface area is:
\[
A_L = 30 \, \text{cm} \times 10 \, \text{cm} = 300 \, \text{cm}^2
\]

#### Step 2: Calculate the area of the hexagonal base
The area of one regular hexagonal base \( A_B \) is given by:
\[
A_B = \frac{3\sqrt{3}}{2} \times \text{side length}^2 = \frac{3\sqrt{3}}{2} \times 5^2 = \frac{3\sqrt{3}}{2} \times 25 = 64.95 \, \text{cm}^2
\]
Since the prism has two hexagonal bases, the total base area is:
\[
2A_B = 2 \times 64.95 = 129.9 \, \text{cm}^2
\]

#### Step 3: Calculate the total surface area
The total surface area \( A_T \) is the sum of the lateral surface area and the areas of the two bases:
\[
A_T = A_L + 2A_B = 300 + 129.9 = 429.9 \, \text{cm}^2
\]

So, the total surface area \( A_T \) is approximately \( 429.9 \, \text{cm}^2 \).

Would you like further explanation or additional details on this calculation?

Here are 5 related questions you might find interesting:
1. How is the volume of this hexagonal prism calculated?
2. How does the surface area formula change if the base is a different polygon?
3. What is the relationship between the perimeter and area of a regular hexagon?
4. How do you derive the area formula for a regular hexagon?
5. How does increasing the height of the prism affect its surface area?

**Tip:** When calculating the area of a regular polygon, always break it down into simpler geometric shapes like triangles.

Find the total surface area of a hexagonal prism with a height of 10 cm, a side length of the hexagonal base of 5 cm, and a width across the flats of 6 cm.

Learn how to calculate the total surface area of a hexagonal prism with a height of 10 cm and a side length of the hexagonal base of 5 cm. This problem covers essential geometry concepts and formulas for finding the surface area of prisms. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation