https://file.aiquickdraw.com/mathbot/file/1727453512497a8e9rway.jpg

What is the volume of the prism, in cubic centimeters?

The volume is 
 cubic centimeters.

What is the surface area of the prism, in square centimeters?

The surface area is 
 square centimeters.

The image you provided shows a triangular prism. To solve for both the **volume** and **surface area** of the prism, let's break down the steps.

### Volume of the Triangular Prism:
The volume \( V \) of a prism is given by the formula:

\[
V = B \times h
\]

Where:
- \( B \) is the area of the triangular base.
- \( h \) is the height (or length) of the prism.

#### Step 1: Find the area of the triangular base:
The triangle's dimensions are given in the image:
- One side is 10 cm.
- The height corresponding to this side is 4 cm.

So, the area \( B \) of the triangular base is:

\[
B = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \, \text{cm} \times 4 \, \text{cm} = 20 \, \text{cm}^2
\]

#### Step 2: Determine the height (length) of the prism:
From the image, the height of the prism is 6 cm.

Now, the volume of the prism can be calculated as:

\[
V = 20 \, \text{cm}^2 \times 6 \, \text{cm} = 120 \, \text{cm}^3
\]

So, the volume of the prism is **120 cubic centimeters**.

### Surface Area of the Triangular Prism:
The surface area \( SA \) of a prism is the sum of the areas of all the faces.

- Two triangular faces (area \( B \) of each triangle is 20 cm²).
- Three rectangular faces.

#### Step 1: Calculate the areas of the rectangular faces:
- The first rectangle has a width of 10 cm (triangle's base) and a height of 6 cm (length of the prism), so its area is:

\[
10 \, \text{cm} \times 6 \, \text{cm} = 60 \, \text{cm}^2
\]

- The second rectangle has a width of 5 cm (side of the triangle) and a height of 6 cm (length of the prism), so its area is:

\[
5 \, \text{cm} \times 6 \, \text{cm} = 30 \, \text{cm}^2
\]

- The third rectangle has a width of 5 cm (the other side of the triangle) and a height of 6 cm (length of the prism), so its area is:

\[
5 \, \text{cm} \times 6 \, \text{cm} = 30 \, \text{cm}^2
\]

#### Step 2: Sum the areas:
- Two triangular faces: \( 2 \times 20 \, \text{cm}^2 = 40 \, \text{cm}^2 \)
- Three rectangular faces: \( 60 \, \text{cm}^2 + 30 \, \text{cm}^2 + 30 \, \text{cm}^2 = 120 \, \text{cm}^2 \)

Now, the total surface area is:

\[
SA = 40 \, \text{cm}^2 + 120 \, \text{cm}^2 = 160 \, \text{cm}^2
\]

So, the surface area of the prism is **160 square centimeters**.

---

Would you like more details or have any questions about the solution?

Here are 5 related questions for practice:
1. What is the volume of a triangular prism with a base area of 25 cm² and a height of 8 cm?
2. How would the surface area change if the length of the prism were doubled?
3. What is the formula for the volume of a rectangular prism?
4. How do you find the lateral surface area of a prism?
5. How would you calculate the volume of a cylinder if its base area and height are known?

**Tip:** When calculating the surface area of a prism, always ensure that you account for both the bases and the lateral faces.

What is the volume of the prism, in cubic centimeters? What is the surface area of the prism, in square centimeters?

Learn how to calculate the volume and surface area of a triangular prism with a base of 10 cm and height 4 cm. The prism has a length of 6 cm. The solution uses geometry concepts and formulas for prism calculations. Suitable for students in Grades 8-10.

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