1. 11.5 Explore Solids & 11.6 Volume of Prisms and Cylinders
You will identify solids
You will find volumes of prisms and cylinders
Essential Questions:
When is a solid a polyhedron?
How do you find the volume of a right prism of right cylinder?

2. EXAMPLE 1
Identify and name polyhedra
Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges. a. b.

3. EXAMPLE 1
Identify and name polyhedron c.
SOLUTION
The solid is formed by polygons, so it is a polyhedron. The two bases are congruent rectangles, so it is a rectangular prism. It has 6 faces, 8 vertices, and 12 edges. a.

4. EXAMPLE 1
Identify and name polyhedron
The solid is formed by polygons, so it is a polyhedron. The base is a hexagon, so it is a hexagonal pyramid. It has 7 faces, consisting of 1 base, 3 visible triangular faces, and 3 non-visible triangular faces. The polyhedron has 7 faces, 7 vertices, and 12 edges. b.
The cone has a curved surface, so it is not a
polyhedron. c.

5. GUIDED PRACTICE
for Example 1
Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges. 1.
The solid is formed by polygons so it is a polyhedron. The base is a square it has 5 faces, 5 vertices and 8 edges.
ANSWER

6. GUIDED PRACTICE
for Example 1 2.
It has a curved square, so it is not a polyhedron.
ANSWER

7. GUIDED PRACTICE
for Example 1 3.
The solid is formed by polygons, so it is a polyhedron. The base is a triangle, so it is a triangular prism. It has 5 face, 6 vertices, and 9 edges.
ANSWER

8. EXAMPLE 2
Use Euler’s Theorem in a real-world situation
House Construction
Find the number of edges on the frame of the house.
SOLUTION
The frame has one face as its
foundation, four that make up its walls, and two that make up its roof, for a total of 7 faces.

9. Use Euler’s Theorem in a real-world situation
EXAMPLE 2
Use Euler’s Theorem in a real-world situation
To find the number of vertices, notice that there are 5 vertices around each pentagonal wall, and there are no other vertices. So, the frame of the house has 10 vertices.
Use Euler’s Theorem to find the number of edges.
F + V = E + 2
= E + 2
15 = E
Euler’s Theorem
Substitute known values.
Solve for E.
The frame of the house has 15 edges.
ANSWER

10. Use Euler’s Theorem with Platonic solids
EXAMPLE 3
Use Euler’s Theorem with Platonic solids
Find the number of faces, vertices, and edges of the regular octahedron.
Check your answer using Euler’s Theorem.
SOLUTION
By counting on the diagram, the octahedron has 8 faces, 6 vertices, and 12 edges. Use Euler’s Theorem to check.
F + V = E + 2 =
14 = 14
Euler’s Theorem
Substitute.
This is a true statement. So, the solution checks.

11. EXAMPLE 4
Describe cross sections
Describe the shape formed by the intersection of the
plane and the cube. a. b.
SOLUTION
a. The cross section is a square.
b. The cross section is a rectangle.

12. EXAMPLE 4
Describe cross sections c.
SOLUTION
c. The cross section is a trapezoid.

13. GUIDED PRACTICE for Examples 2, 3, and 4
4. Find the number of faces, vertices, and edges of the regular dodecahedron on page 796. Check your answer using Euler’s Theorem.
SOLUTION
Counting on the diagram, the dodecahedron has 12 faces, 20 vertices, and 30 edges. Use Euler’s theorem to
F + V = E + 2 =
32 = 32
Check
Euler’s theorem
Substitute
This is a true statement so, the solution check

14. GUIDED PRACTICE
for Examples 2, 3, and 4
Describe the shape formed by the intersection of the
plane and the solid. 5.
ANSWER
The cross section is a triangle

15. GUIDED PRACTICE
for Examples 2, 3, and 4 6.
ANSWER
The cross section is a circle

16. GUIDED PRACTICE
for Examples 2, 3, and 4 7.
ANSWER
The cross section is a hexagon

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

18. 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.

19. 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.
V = Bh = 30(5) = 150cm3

20. EXAMPLE 2
Find volumes of prisms and cylinders
SOLUTION
b. Right cylinder
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

21. 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.

22. 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

23. GUIDED PRACTICE
for Example 1,2,and 3
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

24. 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

25. 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.

26. Find the volume of an oblique cylinder
EXAMPLE 4
Find the volume of an oblique cylinder ≈
Use a calculator.
The volume of the oblique cylinder is about cm3.
ANSWER

27. 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.

28. Solve a real-world problem
EXAMPLE 5
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
= – 4( )
= 35,100 cm2

29. Solve a real-world problem EXAMPLE 5
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.

30. 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

31. GUIDED PRACTICE
for Example 4 and 5
5. Find the volume of the solid shown below.
ANSWER
ft3

32. Daily Homework Quiz
Determine whether the solid is a polyhedron. If it is, name it. 1.
ANSWER no

33. Daily Homework Quiz 2.
ANSWER
yes; pyramid

34. Daily Homework Quiz 3.
ANSWER
yes; pentagonal prism

35. Daily Homework Quiz
4. Find the number of faces, vertices, and edges
of each polyhedron in Exercises 1–3.
ANSWER
pyramid: 5 faces, 5 vertices, 8 edges;
prism: 7 faces, 10 vertices, 15 edges
5. A plane intersects a cone, but does not
intersect the base of the cone. Describe the
possible cross sections.
ANSWER
a point, a circle, an ellipse

36. 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

37. Daily Homework Quiz
3. Find the volume of a right cylinder with height
30 ft and diameter 14 ft.
ANSWER
in.3
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?

38. Daily Homework Quiz
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.

39. When is a solid a polyhedron?
Essential Questions:
When is a solid a polyhedron?
How do you find the volume of a right prism of right cylinder?
You will identify solids
You will find volumes of prisms and cylinders
• A solid is a polyhedron if it is
bounded by polygons. A polyhedron
is regular if all of its faces
are congruent regular polygons.
• For a polyhedron, F + V =E + 2.
• The intersection of a plane and a
solid is a cross section.
If all the faces of a solid are polygons, then the solid is a polyhedron.
• For a prism, V = Bh.
• For a cylinder, V = Bh = π r 2h.
• Cavalieri’s Principle says the
volume formulas work for both
right and oblique prisms and
cylinders.
The volume of a right prism or
right cylinder is the product of the height and the area of the base.