1. MG1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area of base × height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid.
Also covered: AF3.1, AF3.2
California
Standards

2. Vocabulary
Volume
Volume of Prisms Formula

3. Volume is the number of cubic units needed to fill a space.

4. You need 10, or 5 · 2, centimeter cubes to cover the bottom of this rectangular prism.
You need 3 layers of 10 cubes each to fill the prism. It takes 30, or 5 · 2 · 3, cubes.

5. Volume is expressed in cubic units, so the volume of the prism is 5 cm · 2 cm · 3 cm = 30 cubic centimeters, or 30 cm3.

6. The volume of a rectangular prism is the area of its base times its height. This formula can be used to find the volume of any prism.

7. Teacher Example 1: Finding the Volume of a Rectangular Prism
Find the volume of the rectangular prism.
Step 1: Find the area of the base.
13 in.
11 in.
26 in.
B = 26 · 11
The base is a rectangle.
B = 286
Multiply.

8. Teacher Example 1 Continued:
Find the volume of the rectangular prism.
Step 2: Find the volume.
13 in.
11 in.
26 in.
V = Bh
Write the formula.
V = 286 • 13
Substitute for B and h.
V = 3,718 in3
Multiply.
The volume of the prism is 3,718 in3.

9. Student Practice 1: Find the volume of the rectangular prism.
Step 1: Find the area of the base.
16 in.
12 in.
29 in.
B = 29 · 12
The base is a rectangle.
B = 348
Multiply.

10. Student Practice 1 Continued: Find the volume of the rectangular prism.
Step 2: Find the volume.
16 in.
12 in.
29 in.
V = Bh
Write the formula.
V = 348 • 16
Substitute for B and h.
V = 5,568 in3
Multiply.
The volume of the prism is 5,568 in3.

11. You can also use the formula V= Bh to find the volume of a triangular prism. For triangular prisms, B represents the area of a triangle, rather than a rectangle.

12. Teacher Example 2: Finding the Volume of a Triangular Prism
Find the volume of each triangular prism.
A. Step 1: Find the area of the base.
B = ( • 3.9 • 1.3) 1 2 __
The base is a triangle.
B = 2.535
Multiply.

13. Teacher Example 2A: Finding the Volume of a Triangular Prism Continued
Find the volume of each triangular prism.
A. Step 2: Find the volume.
V = Bh
Write the formula.
V = • 4
Substitute for B and h.
V = m3
Multiply.
The volume of the prism is m3.

14. The height of a prism is the distance between its two bases.
Caution!

15. Teacher Example 2B: Finding the Volume of a Triangular Prism
Find the volume of each triangular prism.
B. Step 1: Find the area of the base.
B = ( • 6.5 • 7) 1 2 __
The base is a triangle.
B = 22.75
Multiply.

16. Teacher Example 2B: Finding the Volume of a Triangular Prism Continued
Find the volume of each triangular prism.
B. Step 2: Find the volume.
V = Bh
Write the formula.
V = • 6
Substitute for B and h.
V = ft3
Multiply.
The volume of the prism is ft3.

17. Student Practice 2A:
Find the volume of each triangular prism.
A. Step 1: Find the area of the base.
7 m
1.6 m
4.2 m
B = ( • 4.2 • 1.6) 1 2 __
The base is a triangle.
B = 3.36
Multiply.

18. Find the volume of each triangular prism.
Student Practice 2A Continued:
Find the volume of each triangular prism.
A. Step 2: Find the volume.
7 m
1.6 m
4.2 m
V = Bh
Write the formula.
V = 3.36 • 7
Substitute for B and h.
V = m3
Multiply.
The volume of the prism is m3.

19. Student Practice 2B:
Find the volume of each triangular prism.
B. Step 1: Find the area of the base.
9 ft
5 ft
4.5 ft
B = ( • 4.5 • 9) 1 2 __
The base is a triangle.
B = 20.25
Multiply.

20. Student Practice 2B Continued: Find the volume of each triangular prism.
B. Step 2: Find the volume.
9 ft
5 ft
V = Bh
4.5 ft
Write the formula.
V = • 5
Substitute for B and h.
V = ft3
Multiply.
The volume of the prism is ft3.

21. Teacher Example 3: Application
An artist wants to make glass paper-weights with the dimensions shown. He estimates that he will need less than 20 cubic centimeters of glass for each paperweight. Is his estimate reasonable? Explain.
5.2 cm
3 cm
Step 1: Find the area of the base.
6 cm
B = • 6 • 5.2 = 15.6 1 2 __
The base is a triangle.

22. Teacher Example 3 Continued
An artist wants to make glass paper-weights with the dimensions shown. He estimates that he will need less than 20 cubic centimeters of glass for each paperweight. Is his estimate reasonable? Explain.
5.2 cm
3 cm
Step 2: Find the volume.
6 cm
V = Bh
Write the formula.
V = 15.6 • 3 = 46.8 cm3
Substitute for B and h.
No; each paperweight will require about 47 cm3 of glass.

23. Step 1: Find the area of the base.
Student Practice 3:
An architect wants to make a model building with the dimensions shown. He estimates that he will need more than 60 cubic centimeters of paper for each building. Is his estimate reasonable? Explain.
5.5 cm
5 cm
Step 1: Find the area of the base.
6 cm
B = • 6 • 5.5 = 16.5 1 2 __
The base is a triangle.

24. Student Practice 3 Continued:
An architect wants to make a model building with the dimensions shown. He estimates that he will need more than 60 cubic centimeters of paper for each building. Is his estimate reasonable? Explain.
5.5 cm
5 cm
Step 2: Find the volume.
6 cm
V = Bh
Write the formula.
V = 16.5 • 5 = 82.5 cm3
Substitute for B and h.
No; each building will require about 83 cm3 of paper.

25. 10.8 Warm-Up
Find the volume of each figure.
1. triangular prism with a height of 12 cm and a triangular base with base length 7.3 cm and height 3.5 cm
2. Find the volume of the figure shown.