https://file.aiquickdraw.com/mathbot/file/1729014337700mqq51kix.jpg

The shape in the image appears to be a trapezoidal prism. To find the volume and surface area of this shape, we will need to calculate based on the dimensions provided.

### Given:
- The trapezoid has two parallel sides: one is 12 m, and the other is 6 m.
- The height of the trapezoid is 8 m.
- The length of the prism (depth or distance between the trapezoidal faces) is 10 m.

### Volume of the Trapezoidal Prism:
The formula for the volume \(V\) of a trapezoidal prism is:

\[
V = \text{Area of the trapezoid} \times \text{length (depth)}
\]

The area \(A\) of the trapezoid is given by:

\[
A = \frac{1}{2} \times (b_1 + b_2) \times h
\]

where:
- \(b_1 = 12\) m (larger base)
- \(b_2 = 6\) m (smaller base)
- \(h = 8\) m (height of the trapezoid)

First, calculate the area of the trapezoid:

\[
A = \frac{1}{2} \times (12 + 6) \times 8 = \frac{1}{2} \times 18 \times 8 = 72 \, \text{m}^2
\]

Now, multiply this by the length of the prism (10 m):

\[
V = 72 \times 10 = 720 \, \text{m}^3
\]

So, the volume of the trapezoidal prism is **720 cubic meters**.

### Surface Area of the Trapezoidal Prism:
The surface area \(SA\) of a prism consists of the areas of all its faces. For a trapezoidal prism, this includes:
- 2 trapezoidal faces
- 2 rectangular faces from the non-parallel sides of the trapezoid
- 2 rectangular faces from the parallel sides (bases) of the trapezoid

#### Trapezoidal Faces:
We already know the area of one trapezoidal face is 72 m², and there are two of them:

\[
\text{Area of trapezoidal faces} = 2 \times 72 = 144 \, \text{m}^2
\]

#### Rectangular Faces from Non-Parallel Sides:
The height of the trapezoid (8 m) forms one side of these rectangles, and the length of the prism is 10 m. The two non-parallel sides are slant heights that can be calculated as follows:

- The difference in base lengths is \(12 - 6 = 6\) m, and half of this is \(3\) m, forming one leg of a right triangle with the trapezoid height (8 m). Using the Pythagorean theorem to find the slant height \(s\):

\[
s = \sqrt{8^2 + 3^2} = \sqrt{64 + 9} = \sqrt{73} \approx 8.54 \, \text{m}
\]

Now, calculate the areas of the two rectangular faces formed by the non-parallel sides:

\[
\text{Area of non-parallel faces} = 2 \times 8.54 \times 10 = 170.8 \, \text{m}^2
\]

#### Rectangular Faces from Parallel Sides:
These two faces are simply the areas of the larger base (12 m) and smaller base (6 m) multiplied by the length (10 m):

\[
\text{Area of larger base face} = 12 \times 10 = 120 \, \text{m}^2
\]
\[
\text{Area of smaller base face} = 6 \times 10 = 60 \, \text{m}^2
\]

#### Total Surface Area:
Now, sum all the areas:

\[
SA = 144 + 170.8 + 120 + 60 = 494.8 \, \text{m}^2
\]

So, the total surface area of the trapezoidal prism is **494.8 square meters**.

Would you like further clarification or details on any part of this? Here are five related questions that might interest you:

1. How would the volume change if the height of the trapezoid was different?
2. What would the surface area be if the prism had a different length (depth)?
3. How can you calculate the lateral surface area only, without the bases?
4. What are some real-world applications of calculating the volume and surface area of a trapezoidal prism?
5. How would the volume formula change if the shape was a cylinder instead of a prism?

**Tip:** When calculating the surface area of 3D shapes, breaking it down by each face helps to ensure accuracy.

Learn how to calculate the volume and surface area of a trapezoidal prism with base lengths of 12m and 6m, height of 8m, and depth of 10m. Step-by-step breakdown of key geometry concepts, including the trapezoid area formula and surface area calculation.

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