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.