1. Ch 12 Surface Area and Volume of Solids

2. 12.1 Explore Solids Vocabulary
_____________ - solid that is bounded by polygons, called _______, that enclose a single region of space.
POLYHEDRON
FACES
FACE
EDGE
VERTEX

3. TYPES OF SOLIDS POLYHEDRONS NOT POLYHEDRON PRISMS CYLINDERS PYRAMIDS
SPHERES
CONES

4. Modeled Ex: 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.
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.
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.

5. Modeled Ex : 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. c.
The cone has a curved surface, so it is not a
polyhedron. c.

6. Guided Ex : Your Turn (Partner Practice)
Tell whether the solid is a polyhedron. If it is, name the polyhedron and find the number of faces, vertices, and edges. 1. 2. 3.
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
2. It has a curved square, so it is not a polyhedron.
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.

7. Euler’s Theorem
The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula
F + V = E + 2
F = 6, V = 8, E = 12
F + V = E + 2
6 + 8 =

8. Modeled Ex: Use Euler’s Theorem in real-world situation
Find the number of edges on the frame of the house.
SOLUTION
The frame has one face as a total of 7 faces.
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

9. Modeled Ex: Describe cross sections
Describe the shape formed by the intersection of the
plane and the solid. a. b.
SOLUTION
a. The cross section is a square.
b. The cross section is a rectangle.

10. Assignment
HW 12.1
Pg 798 #3 – 20, 25-27, 31.

11. Ch 12 Outlook Date Section Topic April 12/13  12.1 Explore Solids
12.2, 12.3
Surface Area of Prisms, Cylinders, Pyramids, and Cones
April 18/19
12.4, 12.5
Quiz #1 ( )
Volume of Prisms, Cylinders, Pyramids, and Cones
April 20/21
12.6
Quiz #2 ( )
Surface Area and Volume of Spheres
April 22/25
12.7
Explore Similar Solids
April 26/27
Review
Quiz #3 ( )
April 28/29
Chapter 12 Test
Test

12. 12.2 Surface Area Prisms and Cylinders
VOCABULARY
A _______is a polyhedron with two congruent faces, called bases, that lie in parallel planes.
The ____________of a prism are parallelograms formed by connecting the corresponding vertices of the bases.
The _____________of a prism are the segments connecting the corresponding vertices of the bases.
OCTAGONAL PRISM
prism
lateral faces
lateral edges

13. The ____________of a polyhedron is the sum of the areas of its faces.
PENTAGONAL PRISM
VOCABULARY
The ____________of a polyhedron is the sum of the areas of its faces.
The __________ of a polyhedron is the sum of the areas of its lateral faces.
surfaces area
lateral area

14. CLICK FOR EXAMPLES OF NETS
A ____is a two dimensional representation of the faces of a polyhedron.
NET
CLICK FOR EXAMPLES OF NETS

15. PRISMS RIGHT PRISMS OBLIQUE PRISMS
In a right prism, each lateral edge is perpendicular to both bases.
An oblique prism is a prism with lateral edges that are not perpendicular bases.

16. EXAMPLE 1
Find the surface area of a rectangular prism with height 2 centimeters, length 5 centimeters, and width 6 centimeters.
STEP 1
SOLUTION
Sketch the prism. Imagine unfolding it to
make a net.

17. Find the areas of the rectangles that form the faces of the prism.
STEP 2
STEP 3
Add the areas of all the faces to find the
surface area.
The surface area of the prism is
S = 2(12) + 2(10) + 2(30)
= 104 cm2.

18. EXAMPLE 2
Find the surface area of a right prism
Find the surface area of the right pentagonal prism.
SOLUTION
STEP 1
Find the perimeter and area of a base of the prism.
Each base is a regular pentagon.

19. STEP 2
Use the formula for the surface area that
uses the apothem.
S = aP + Ph
≈ (4.86)(35.25) + (35.25)(9) ≈
The surface area of the right pentagonal prism is
about square feet.
ANSWER

20. Find the height of a cylinder
EXAMPLE 3
Find the height of a cylinder
COMPACT DISCS
You are wrapping a stack of 20 compact discs using a shrink wrap. Each disc is cylindrical with height 1.2 millimeters and radius 60 millimeters. What is the minimum amount of shrink wrap needed to cover the stack of 20 discs?
SOLUTION
The 20 discs are stacked, so the height of the stack will be 20(1.2) = 24 mm. The radius is 60 millimeters. The minimum amount of shrink wrap needed will be equal to the surface area of the stack of discs.
S = 2πr πrh
= 2π(60) π(60)(24)
≈ 31,667
You will need at least 31,667 square millimeters, or about 317 square centimeters of shrink wrap.
ANSWER

21. EXAMPLE 4
Find the height of a cylinder
Find the height of the right cylinder shown, which has a surface area of square meters.
SOLUTION
Substitute known values in the formula for the surface area of a right cylinder and solve for the height h.
S = 2πr2 + 2πrh
= 2π(2.5)2 + 2π(2.5)h
The height of the cylinder is about 7.5 m.
ANSWER

22. 12.3 Surface Area of Pyramids, & Cones
VOCABULARY
A ____________________ is the common vertex of the lateral faces of a pyramid.
The ____________ of a regular pyramid is the height of a face of the regular pyramid.
PENTAGONAL PRISM
vertex of a pyramid
slant height

23. Find the area of a lateral face of a pyramid
EXAMPLE 1
Find the area of a lateral face of a pyramid
A regular square pyramid has a height of 15 centimeters and a base edge length of 16 centimeters. Find the area of each lateral face of the pyramid
SOLUTION
A = bl = (16)(17)
= 136 square centimeters. 1 2
The area of each triangular face is
ANSWER

24. EXAMPLE 2
Find the surface area of a pyramid
Find the surface area of the regular hexagonal pyramid.
SOLUTION
The surface area of the regular hexagonal pyramid is about ft2.
ANSWER

25. Standardized Test Practice
EXAMPLE 3
Standardized Test Practice
SOLUTION
Find the slant height l of the right cone
Find the surface area of a right cone.
l2 = h2+ r 2
S = πr2 + πrl
= π(62) + π(6)(10)
l2 =
l = 10
= 96π
The correct answer is B.
ANSWER

26. EXAMPLE 4
Find the lateral area of a cone
TRAFFIC CONE
The traffic cone can be approximated by a right cone with radius 5.7 inches and height 18 inches. Find the approximate lateral area of the traffic cone.
SOLUTION
To find the slant height l, use the Pythagorean Theorem.
l2 = (5.7)2, so l ≈ inches.
Find the lateral area.
The lateral area of the traffic cone is about 338.4
square inches.
ANSWER
Lateral area = πrl
= π(5.7)(18.9) ≈

27. Assignment HW 12.2 Pg 806 #4-12 Even, 13-20 HW 12.3
Pg 814 #3-19, Odd, 22-24
QUIZ #1 (12.1 – 12.3) NEXT TIME

28. Ch 12 Outlook Date Section Topic April 12/13  12.1 Explore Solids
12.2, 12.3
Surface Area of Prisms, Cylinders, Pyramids, and Cones
April 18/19
12.4, 12.5
Quiz #1 ( )
Volume of Prisms, Cylinders, Pyramids, and Cones
April 20/21
12.6
Quiz #2 ( )
Surface Area and Volume of Spheres
April 22/25
12.7
Explore Similar Solids
April 26/27
Review
Quiz #3 ( )
April 28/29
Chapter 12 Test
Test

29. 12.4 Volume of Prisms and Cylinders
RECTANGULAR
PRISM
VOCABULARY
The _________ of a solid is the number of cubic units contained in its interior. 2
units
volume 4
units 2
units

30. POSTULATES V = s3 CUBE The volume of a cube is _______
Volume Congruence Postulate
If two polyhedra are congruent, then _____________________.
Volume Addition Postulate
The volume of a solid is the _____ of the volumes of all its nonoverlapping parts.
V = s3
they have the same volume
sum

31. Find the number of unit cubes
EXAMPLE 1
Find the number of unit cubes
3- D PUZZLE
Find the volume of the puzzle piece
in cubic units.
SOLUTION
By the Volume Addition Postulate, the total volume of the puzzle piece is
= 20 cubic units.

32. Theorems V = (B)H V = (B)H or V=( r2) H VOLUME OF A PRISM
The volume of any right prism or right cylinder can be found by multiplying (AREA OF BASE) x HEIGHT or
VOLUME OF A CYLINDER
The volume of any right prism or right cylinder can be found by multiplying AREA OF BASE x HEIGHT or
V = (B)H
V = (B)H or V=( r2) H

33. EXAMPLE 2
Find volumes of prisms and cylinders
Find the volume of the solid.
a. Right trapezoidal prism
b. Right cylinder
SOLUTION
(3)(6 + 14)
a. The area of a base is 1 2
= 30cm2 and h
= 5 cm.
V = Bh = 30(5) = 150cm3
b. The area of the base is
π 92, or 81πft2. Use h = 6 ft.
V = Bh = 81π(6) = 486π ≈ ft3

34. 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
90 in3. = x3
4.48 in. ≈ x

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

36. 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
= π(42)(7)
= 112π ≈
The volume of the oblique cylinder is about cm3.
ANSWER

37. EXAMPLE 5
Solve a real-world problem
SCULPTURE
The sculpture is made up of 13 beams. In centimeters, suppose the dimensions of each beam are 30 by 30 by 90. Find its volume.
ANSWER
The volume of the sculpture is 1,053,000 cm3, or m3.

38. 12.5 Volume of Pyramids and Cones
Theorems
VOLUME OF A PYRAMID
The volume V of a pyramid is (AREA OF BASE) x height or
VOLUME OF A CONE
The volume V of a cone is 1/3 (AREA OF BASE) x height or

39. Find the volume of a solid
EXAMPLE 1
Find the volume of a solid
Find the volume of the solid. b. a.
V = Bh 1 3
V = Bh 1 3
= ( )(9) 1 3 2
= (πr2)h 1 3
= (π )(4.5) 1 3
= 36 m3
= 7.26π
≈ cm3

40. EXAMPLE 2
Use volume of a pyramid
ALGEBRA
Originally, the pyramid had height 144 meters and volume 2,226,450 cubic meters. Find the side length of the square base.
SOLUTION
V = bh 1 3
2,226,450 = (x2)(144) 1 3
6,679,350 = 144x2
46,384 ≈ x2
Originally, the side length of the base was about 215 meters.
ANSWER
215 ≈ x

41. EXAMPLE 3
Use trigonometry to find the volume of a cone
Find the volume of the right cone.
SOLUTION
To find the radius r of the base,
use trigonometry.
tan 65° =
opp.
Adj.
tan 65° = 16 r
r =
tan 65° 16
≈ 7.46
ANSWER
Use the formula for the volume of a cone.
V = (π r 2)h ≈ π(7.462)(16) ≈ ft3 3 1

42. EXAMPLE 4
Find volume of a composite solid
Find the volume of the solid shown.
SOLUTION
= s Bh 1 3
= (6)2 6 1 3 =
= 288
The volume of the solid is 288 cubic meters.
ANSWER

43. Assignment
HW 12.4
Pg 823 #4, 7-12, 18-20, 22, 25
HW 12.5
Pg 832 #3-8, Even

44. Ch 12 Outlook Date Section Topic April 12/13  12.1 Explore Solids
12.2, 12.3
Surface Area of Prisms, Cylinders, Pyramids, and Cones
April 18/19
12.4, 12.5
Quiz #1 ( )
Volume of Prisms, Cylinders, Pyramids, and Cones
April 20/21
12.6
Quiz #2 ( )
Surface Area and Volume of Spheres
April 22/25
12.7
Explore Similar Solids
April 26/27
Review
Quiz #3 ( )
April 28/29
Chapter 12 Test
Test

45. 12.6 Surface Area and Volume of Spheres

47. 12.7 Explore Similar Solids