1. Find the area of each polygon or circle.
1. Trapezoid, bases 12 ft and 18 ft, height 3 ft
ANSWER
324 ft2
2. Circle, diameter 8.2 in.
ANSWER
7.27 in.2
3. Regular hexagon, side length 14 cm
ANSWER
cm2

2. Vocabulary
volume – of a solid is the number of cubic units contained in its interior.
Volume Congruence Postulate 28 – If two polyhedra are congruent, they have the same volume.
Volume Addition Postulate 29 – The volume of a solid is the sum of the volumes of all its nonoverlapping parts.

3. Vocabulary
Volume of a Cube Postulate 27 –
V = s3
Volume of a Prism Theorem 11.6 – V = Bh
Volume of a Cylinder Theorem 11.7 – V = Bh = πr 2h
where B = base area, h = height, r = radius, s = side

4. Vocabulary
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.

5. EXAMPLE 1
Find the number of unit cubes
3- D PUZZLE
Find the volume of the puzzle piece in cubic units.

6. EXAMPLE 1
Find the number of unit cubes
SOLUTION
To find the volume, find the number of unit cubes it contains. Separate the piece into three rectangular boxes as follows:
The base is 7 units by 2 units. So, it contains 7 . 2, or 14 unit cubes. The upper left box is 2 units by 2 units. So, it contains 2 . 2, or 4 unit cubes. The upper right box is 1 unit by 2 units. So, it contains 1 . 2, or 2 unit cubes.
By the Volume Addition Postulate, the total volume of the puzzle piece is
= 20 cubic units.

7. EXAMPLE 2
Find volumes of prisms and cylinders
Find the volume of the solid.
a. Right trapezoidal prism
SOLUTION
(3)(6 + 14)
a. The area of a base is 1 2
= 30cm2 and h
= 5 cm.
Use h = 5 cm to find the volume.
V = Bh = 30(5) = 150cm3

8. EXAMPLE 2
Find volumes of prisms and cylinders
Find the volume of the solid.
b. Right cylinder
SOLUTION
b. The area of the base is
π 92, or 81πft2. Use h = 6 ft to find the volume.
V = Bh = 81π(6) = 486π ≈ ft3

9. The volume of the cube is 90 cubic inches. Find the value of x.
EXAMPLE 3
Use volume of a prism
ALGEBRA
The volume of the cube is
90 cubic inches. Find the value of x.
SOLUTION
A side length of the cube is x inches.
V = x3
Formula for volume of a cube
90 in3 = x3
Substitute for V.
4.48 in ≈ x
Find the cube root.

10. GUIDED PRACTICE
for Example 1,2,and 3
1. Find the volume of the puzzle
piece shown in cubic units.
The volume of the puzzle cube is 7 units3
ANSWER
2. Find the volume of a square prism that has a base edge length of 5 feet and a height of 12 feet.
The volume of a square prism is 300 ft3
ANSWER

11. GUIDED PRACTICE
for Example 1,2,and 3
3. The volume of a right cylinder is 684π cubic
inches and the height is 18 inches. Find the radius.
The radius of a right cylinder is in 38
ANSWER

12. Find the volume of an oblique cylinder
EXAMPLE 4
Find the volume of an oblique cylinder
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.
V = π r2h
Formula for volume of a cylinder
= π(42)(7)
Substitute known values.
= 112π
Simplify. ≈
Use a calculator.
The volume of the oblique cylinder is about cm3.
ANSWER

13. EXAMPLE 5
Solve a real-world problem
PLANTER
The planter is made up of 13 beams. In centimeters, suppose the dimensions of each beam are 30 by 30 by 90. Find its volume.
SOLUTION
The area of the base B can be found by subtracting the area of the small rectangles from the area of the large rectangle.
B = Area of large rectangle –
4 Area of small rectangle

14. Solve a real-world problem EXAMPLE 5
B = Area of large rectangle – 4 Area of small rectangle
= – 4( )
= 35,100 cm2
Use the formula for the volume of a prism.
V = Bh
Formula for volume of a prism.
= 35,100(30)
Substitute.
= 1,053,000 cm3
Simplify.
The volume of the planter is 1,053,000 cm3, or m3.

15. GUIDED PRACTICE
for Example 4 and 5
4. Find the volume of the oblique prism shown below.
The volume of the oblique prism is 180m3
ANSWER
5. Find the volume of the solid shown below.
ANSWER
ft3

16. Daily Homework Quiz
1. Find the volume of the solid at the right.
ANSWER
in.3
2. Find the volume of a right triangular prism with height 32 in., base height 12 in., and base length 18 in.
ANSWER
3456 in.3
3. Find the volume of a right cylinder with height
30 ft and diameter 14 ft.
ANSWER
in.3

17. Daily Homework Quiz
4. A cylindrical beaker with diameter 10 in. and
height 12 in. is filled with water that is then
poured into a rectangular pan that is 14 in. by
9 in. by 3 in. What is the volume of each solid?
Would the water overflow the pan? If so, what
is the height of the water in the beaker after
the pan is filled?
ANSWER
cylinder: in.3; pan: 378 in.3; Yes, it would
overflow the pan. After the pan is filled , the
height of the water in the beaker will be about
7.2 in.