The image you have provided is a trapezoidal prism. To find the surface area of this trapezoidal prism, we need to calculate the areas of all its faces.

Step 1: Calculate the area of the trapezoidal faces

The trapezoid has a height of 4.5 cm, one base of 5 cm, and the other base of 3 cm.

The formula for the area of a trapezoid is: $A = \frac{1}{2} \times (b_1 + b_2) \times h$ where:

• $b_1 = 5 \, \text{cm}$ (one base),
• $b_2 = 3 \, \text{cm}$ (other base),
• $h = 4.5 \, \text{cm}$ (height).

$A_{\text{trapezoid}} = \frac{1}{2} \times (5 + 3) \times 4.5 = \frac{1}{2} \times 8 \times 4.5 = 18 \, \text{cm}^2$

There are two trapezoidal faces, so the total area of the trapezoidal faces is: $A_{\text{two trapezoids}} = 2 \times 18 = 36 \, \text{cm}^2$

Step 2: Calculate the area of the rectangular faces

There are three rectangular faces:

1. The front and back faces, which have dimensions $11 \, \text{cm} \times 4.5 \, \text{cm}$,
2. The bottom face, which has dimensions $11 \, \text{cm} \times 5 \, \text{cm}$,
3. The top face, which has dimensions $11 \, \text{cm} \times 3 \, \text{cm}$.

Front and back face:

$A_{\text{front/back}} = 11 \times 4.5 = 49.5 \, \text{cm}^2$

Bottom face:

$A_{\text{bottom}} = 11 \times 5 = 55 \, \text{cm}^2$

Top face:

$A_{\text{top}} = 11 \times 3 = 33 \, \text{cm}^2$

Step 3: Total Surface Area

Now, sum the areas of all the faces: $A_{\text{total}} = 36 + 49.5 + 49.5 + 55 + 33$ $A_{\text{total}} = 223 \, \text{cm}^2$

Thus, the total surface area of the trapezoidal prism is 223 cm².

Would you like further details or explanations?

Here are 5 related questions you might want to explore:

1. How do you calculate the volume of this trapezoidal prism?
2. What is the lateral surface area of this trapezoidal prism?
3. How does the surface area change if the height of the prism is doubled?
4. How do the formulas for the surface area differ for other types of prisms?
5. Can the same formula for the trapezoid area be applied in a real-world scenario, such as roof design?

Tip: When dealing with 3D shapes, breaking down the faces into simpler geometric shapes like rectangles and trapezoids helps in calculating the surface area efficiently.