Question

volume & surface area homework (2)
date:
1.)

a.) find the volume. show all work! (find
the area of the base, and multiply by the height)

b.) draw the net or decompose the
triangular prism.

c.) find the surface area of the triangular
prism. show all work!

Explanation:

Part a: Find the Volume

Step 1: Find the area of the triangular base

The base of the triangle is \( 8 \, \text{cm} \) and the height of the triangle is \( 3 \, \text{cm} \). The formula for the area of a triangle is \( A = \frac{1}{2} \times \text{base} \times \text{height} \).
\[
A = \frac{1}{2} \times 8 \times 3 = 12 \, \text{cm}^2
\]

Step 2: Multiply the base area by the length of the prism

The length of the prism (the distance along the direction perpendicular to the triangular base) is \( 10 \, \text{cm} \). The formula for the volume \( V \) of a prism is \( V = \text{Base Area} \times \text{length} \).
\[
V = 12 \times 10 = 120 \, \text{cm}^3
\]

Step 1: Calculate the area of the two triangular bases

We already know the area of one triangular base is \( 12 \, \text{cm}^2 \) (from part a). So the area of two triangular bases is \( 2\times12 = 24 \, \text{cm}^2 \).

Step 2: Calculate the area of the three rectangular faces

• The area of the rectangle with dimensions \( 8 \, \text{cm} \times 10 \, \text{cm} \) is \( A_1=8\times10 = 80 \, \text{cm}^2 \).
• The area of each of the two rectangles with dimensions \( 5 \, \text{cm} \times 10 \, \text{cm} \) is \( A_2 = 5\times10=50 \, \text{cm}^2 \). So the area of two such rectangles is \( 2\times50 = 100 \, \text{cm}^2 \).

Step 3: Sum up the areas of the bases and the lateral faces

The surface area \( SA \) of the triangular prism is the sum of the area of the two triangular bases and the area of the three rectangular faces.
\[
SA=24 + 80+100=204 \, \text{cm}^2
\]

Answer:

The volume of the triangular prism is \( \boldsymbol{120 \, \text{cubic centimeters}} \).

Part b: Draw the Net of the Triangular Prism

A triangular prism has two triangular bases and three rectangular faces.

1. Draw the two triangular bases (each with base \( 8 \, \text{cm} \) and height \( 3 \, \text{cm} \)).
2. Draw the three rectangles:

• One rectangle with dimensions \( 8 \, \text{cm} \times 10 \, \text{cm} \) (attached to the base of the triangle).
• Two rectangles with dimensions \( 5 \, \text{cm} \times 10 \, \text{cm} \) (attached to the other two sides of the triangle).

Arrange the triangles and rectangles such that the rectangles connect the corresponding sides of the two triangles. (Since this is a drawing instruction, a text - based description of the net structure is provided above. For an actual drawing, you would sketch the two triangles and three rectangles as described.)

Part c: Find the Surface Area