Question

name: date: volume & surface area homework (2) 1.) diagram of triangular prism with 5 cm, 10 cm, 3 cm, 8 cm 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: Volume Calculation

Step1: Find area of triangular base

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

Step2: Multiply by length of prism

The length (height of the prism) is \( l = 10 \, \text{cm} \). The volume \( V \) of a prism is \( V = A \times l \).
\[
V = 12 \times 10 = 120 \, \text{cm}^3
\]

A triangular prism's net consists of two congruent triangular bases and three rectangular faces.

1. Draw two triangles (base: \( 8 \, \text{cm} \), height: \( 3 \, \text{cm} \)) separated by some space.
2. Draw three rectangles:

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

1. Arrange the rectangles between the two triangles such that each side of the triangle is connected to a rectangle.

(Note: Since drawing is visual, the description above outlines the components. The net will have two triangles and three rectangles as described.)

Step1: Area of two triangular bases

From Part a, the area of one triangle is \( 12 \, \text{cm}^2 \). For two triangles:
\[
A_{\text{triangles}} = 2 \times 12 = 24 \, \text{cm}^2
\]

Step2: Area of three rectangular faces

• Rectangle 1 (base \( 8 \, \text{cm} \), length \( 10 \, \text{cm} \)): \( A_1 = 8 \times 10 = 80 \, \text{cm}^2 \)
• Rectangle 2 (side \( 5 \, \text{cm} \), length \( 10 \, \text{cm} \)): \( A_2 = 5 \times 10 = 50 \, \text{cm}^2 \)
• Rectangle 3 (side \( 5 \, \text{cm} \), length \( 10 \, \text{cm} \)): \( A_3 = 5 \times 10 = 50 \, \text{cm}^2 \)

Total area of rectangles:
\[
A_{\text{rectangles}} = 80 + 50 + 50 = 180 \, \text{cm}^2
\]

Step3: Total surface area

Add the area of triangles and rectangles:
\[
\text{Surface Area} = 24 + 180 = 204 \, \text{cm}^2
\]

Answer:

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

Part b: Net of Triangular Prism