1. Volume of Prisms and Cylinders
Goal: Find volumes of prisms and Cylinders.

2. Vocabulary
Volume: The volume of a solid is the number of cubic units contained in its interior.

3. Postulate 27: Volume of a Cube Postulate
The volume of a cube is the cube of the length of its side.
V = s³

4. Postulate 28:Volume Congruence Postulate
If two polyhedra are congruent, then they have the same volume.

5. Postulate 29: Volume Addition Postulate
The volume of a solid is the sum of the volumes of all its nonoverlapping parts.

6. Theorem 12.6: Volume of a Prism
The volume of a prism is V = Bh, where B is the area of the base and h is the height.

7. Theorem 12.7: Volume of a Cylinder
The volume V of a cylinder is
V = Bh = πr²h, where B is the area of the base, h is the height, and r is the radius of a base.

8. Example 1(a) Find the volume of the right triangular prism.
The area of the base is ½(1)(3) = 3/2 ft² and h = 2 ft.
V = Bh = 3/2 • 2 = 3 ft³

9. Example 1(b)
The area of the base is π • 5², or 25π m². Use h = 6 to find the volume.
V = Bh = 25π(6) = 150π ≈ m³

10. Example 2
The volume of the right cylinder shown is 1253 cubic centimeters. Find the value of x.
Solution
The area of the base is πx² square centimeters.

11. Example 2 V = Bh Formula for volume of a cylinder
1253 = πx²(10) Substitute
1253 = 10πx² Rewrite
1253/10π = x² Divide each side by 10π
39.88 ≈ x² Simplify
6.32 ≈ x Find the positive square root
The radius of the cylinder is about 6.32 centimeters.

12. Theorem 12.8: Cavalieri’s Principle
If two solids have the same height and the same cross-sectional area at every level, then they have the same volume.

13. Example 3 Find the volume of the oblique cylinder. Solution
Cavalieri’s Principle allows you to use Theorem 12.7 to find the volume of the oblique cylinder.

14. Example 3 V = πr²h Formula for volume of a cylinder
= π(5²)(8) Substitute
= 200π Simplify
≈ Use a calculator
The volume of the oblique cylinder is about in.

15. Example 4 Find the volume of the solid. Solution
The area of the base B can be found by subtracting the area of the small circle from the area of the large circle.
B = Area of large circle – Area of small circle
= π(3²) – π(1²) = 8π ≈ 25.13mm²

16. Example 4 Use the formula for volume of a cylinder.
V = Bh Formula for volume of a cylinder
= (25.13)(7) Substitute
= Use a calculator
The volume of the solid is about cubic millimeters.

17. Checkpoint 1 Find the volume of the right rectangular prism.
220 cubic meters
V = Bh
V = (11)(4)(5)
V = 220

18. Checkpoint 2
Find the volume of the right cylinder.
cubic feet

19. Checkpoint 3
The volume of the right cylinder is 200π cubic centimeters. Find the value of x.
10cm

20. Checkpoint 4 Find the volume of the oblique cylinder.
cubic millimeters

21. Checkpoint 5
Find the volume of the composite solid.
89.13 cubic in.

22. - Soren Aabye Kierkegaard (1813-1855)
"People demand freedom of speech to make up for the freedom of thought which they avoid."
- Soren Aabye Kierkegaard ( )