surface area and volumes

Сентябрь 6, 2022

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surface area and volumes

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2. Polyhedrons What is a polyhedron? Circles are not polygons
3. Identifying Polyhedrons A polyhedron is a solid that is bounded by polygons, called faces, that enclose
4. Parts of a Polyhedron
5. Example 1 Counting Faces, Vertices, and Edges Count the faces, vertices, and edges of each polyhedron
6. Example 1A Counting Faces Count the faces, vertices, and edges of each polyhedron 4 faces
7. Example 1a Counting Vertices Count the faces, vertices, and edges of each polyhedron 4 vertices
8. Example 1a Counting Edges Count the faces, vertices, and edges of each polyhedron 6 edges
9. Example 1b Counting Faces Count the faces, vertices, and edges of each polyhedron 5 faces
10. Example 1b Counting Vertices Count the faces, vertices, and edges of each polyhedron 5 vertices
11. Example 1b Counting Vertices Count the faces, vertices, and edges of each polyhedron 8 edges
12. Example 1c Counting Faces Count the faces, vertices, and edges of each polyhedron 6 faces
13. Example 1c Counting Vertices Count the faces, vertices, and edges of each polyhedron 6 vertices
14. Example 1c Counting Edges Count the faces, vertices, and edges of each polyhedron 10 edges
15. Notice a Pattern?
16. Theorem 12.1 Euler's Theorem The number of faces (F), vertices (V), and edges (E) of a
17. The surface of a polyhedron consists of all points on its faces A polyhedron is convex
18. Regular Polyhedrons A polyhedron is regular if all its faces are congruent regular polygons. regular Not
19. 5 kinds of Regular Polyhedrons 4 faces 6 faces 8 faces 12 faces 20 faces
20. Example 2 Classifying Polyhedrons One of the octahedrons is regular. Which is it? A polyhedron is
21. Example 2 Classifying Polyhedrons All its faces are congruent equilateral triangles, and each vertex is formed
22. Example 3 Counting the Vertices of a Soccer Ball A soccer ball has 32 faces: 20
23. Example 3 Counting the Vertices of a Soccer Ball A soccer ball has 32 faces: 20
24. Prisms A prism is a polyhedron that has two parallel, congruent faces called bases. The other
25. Prisms The altitude or height, of a prism is the perpendicular distance between its bases In
26. Surface Area of a Prism The surface area of a polyhedron is the sum of the
27. Example 1 Find the Surface Area of a Prism The Skyscraper is 414 meters high. The
28. Example 1 Find the Surface Area of a Prism The Skyscraper is 414 meters high. The
29. Example 1 Find the Surface Area of a Prism The Skyscraper is 414 meters high. The
30. Nets A net is a pattern that can be cut and folded to form a polyhedron.
31. Surface Area of a Right Prism The surface area, S, of a right prism is S
32. Example 2 Finding the Surface Area of a Prism Find the surface area of each right
33. Example 2 Finding the Surface Area of a Prism Find the surface area of each right
34. Example 2 Finding the Surface Area of a Prism Find the surface area of each right
35. Cylinders A cylinder is a solid with congruent circular bases that lie in parallel planes The
36. Surface Area of a Right Cylinder The surface area, S, of a right circular cylinder is
37. Example 3 Finding the Surface Area of a Cylinder Find the surface area of the cylinder
38. Example 3 Finding the Surface Area of a Cylinder Find the surface area of the cylinder
39. Pyramids A pyramid is a polyhedron in which the base is a polygon and the lateral
40. Pyramids The intersection of two lateral faces is a lateral edge The intersection of the base
41. Regular Pyramid A pyramid is regular if its base is a regular polygon and if the
42. Developing the formula for surface area of a regular pyramid Area of each triangle is ½bL
43. Surface Area of a Regular Pyramid The surface area, S, of a regular pyramid is S
44. Example 1 Finding the Surface Area of a Pyramid Find the surface area of each regular
45. Example 1 Finding the Surface Area of a Pyramid Find the surface area of each regular
46. Example 1 Finding the Surface Area of a Pyramid Find the surface area of each regular
47. Cones A cone is a solid that has a circular base and a vertex that is
48. Right Cone A right cone is one in which the vertex lies directly above the center
49. Developing the formula for the surface area of a right cone Use the formula for surface
50. Surface Area of a Right Cone The surface area, S, of a right cone is S
51. Example 2 Finding the Surface Area of a Right Cone Find the surface area of the
52. Example 2 Finding the Surface Area of a Right Cone Find the surface area of the
53. Volume formulas The Volume, V, of a prism is V = Bh The Volume, V, of
54. Volume The volume of a polyhedron is the number of cubic units contained in its interior
55. Postulates All the formulas for the volumes of polyhedrons are based on the following three postulates
56. Example 1: Finding the Volume of a Rectangular Prism The cardboard box is 5" x 3"
57. Example 1: Finding the Volume of a rectangular Prism The cardboard box is 5" x 3"
58. Volume of a Prism and a Cylinder Cavalieri's Principle If two solids have the same height
59. Volume of a Prism The Volume, V, of a prism is V = Bh where B
60. Volume of a Cylinder The volume, V, of a cylinder is V=Bh or V = πr2h
61. Example 2 Finding Volumes Find the volume of the right prism and the right cylinder
62. Example 2 Finding Volumes Find the volume of the right prism and the right cylinder 3
63. Example 2 Finding Volumes Find the volume of the right prism and the right cylinder Area
64. Example 3 Estimating the Cost of Moving You are moving from Newark, New Jersey, to Golden,
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Polyhedrons
What is a polyhedron?
Circles are not polygons

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Identifying Polyhedrons
A polyhedron is a solid that is bounded by polygons,

called faces, that enclose a single region of space.
An edge of polyhedron is a line segment formed by the intersection of two faces
A vertex of a polyhedron is a point where three or more edges meet

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Parts of a Polyhedron

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Example 1 Counting Faces, Vertices, and Edges
Count the faces, vertices, and edges

of each polyhedron

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Example 1A Counting Faces
Count the faces, vertices, and edges of each polyhedron
4

faces

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Example 1a Counting Vertices
Count the faces, vertices, and edges of each polyhedron
4

vertices

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Example 1a Counting Edges
Count the faces, vertices, and edges of each polyhedron
6

edges

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Example 1b Counting Faces
Count the faces, vertices, and edges of each polyhedron
5

faces

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Example 1b Counting Vertices
Count the faces, vertices, and edges of each polyhedron
5

vertices

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Example 1b Counting Vertices
Count the faces, vertices, and edges of each polyhedron
8

edges

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Example 1c Counting Faces
Count the faces, vertices, and edges of each polyhedron
6

faces

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Example 1c Counting Vertices
Count the faces, vertices, and edges of each polyhedron
6

vertices

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Example 1c Counting Edges
Count the faces, vertices, and edges of each polyhedron
10

edges

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Notice a Pattern?

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Theorem 12.1 Euler's Theorem
The number of faces (F), vertices (V), and edges

(E) of a polyhedron is related by F + V = E + 2

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The surface of a polyhedron consists of all points on its

faces
A polyhedron is convex if any two points on its surface can be connected by a line segment that lies entirely inside or on the polyhedron

More about Polyhedrons

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Regular Polyhedrons
A polyhedron is regular if all its faces are congruent

regular polygons.

regular

Not regular
Vertices are not formed by the same number of faces

3 faces

4 faces

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5 kinds of Regular Polyhedrons
4 faces
6 faces
8 faces
12 faces
20 faces

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Example 2 Classifying Polyhedrons
One of the octahedrons is regular. Which is it?
A

polyhedron is regular if all its faces are congruent regular polygons.

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Example 2 Classifying Polyhedrons
All its faces are congruent equilateral triangles, and each

vertex is formed by the intersection of 4 faces

Faces are not all congruent
(regular hexagons and squares)

Faces are not all regular polygons or congruent
(trapezoids and triangles)

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Example 3 Counting the Vertices of a Soccer Ball
A soccer ball has

32 faces: 20 are regular hexagons and 12 are regular pentagons. How many vertices does it have?
A soccer ball is an example of a semiregular polyhedron - one whose faces are more than one type of regular polygon and whose vertices are all exactly the same

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Example 3 Counting the Vertices of a Soccer Ball
A soccer ball has

32 faces: 20 are regular hexagons and 12 are regular pentagons. How many vertices does it have?
Hexagon = 6 sides, Pentagon = 5 sides
Each edge of the soccer ball is shared by two sides
Total number of edges = ½(6●20 + 5●12)= ½(180)= 90
Now use Euler's Theorem
F + V = E + 2
32 + V = 90 + 2
V = 60

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Prisms
A prism is a polyhedron that has two parallel, congruent faces

called bases.
The other faces, called lateral faces, are parallelograms and are formed by connecting corresponding vertices of the bases
The segment connecting these corresponding vertices are lateral edges
Prisms are classified by their bases

base

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Prisms
The altitude or height, of a prism is the perpendicular distance

between its bases
In a right prism, each lateral edge is perpendicular to both bases
Prisms that have lateral edges that are oblique (≠90°) to the bases are oblique prisms
The length of the oblique lateral edges is the slant height of the prism

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Surface Area of a Prism
The surface area of a polyhedron

is the sum of the areas of its faces

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Example 1 Find the Surface Area of a Prism
The Skyscraper is 414

meters high. The base is a square with sides that are 64 meters. What is the surface area of the skyscraper?

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Example 1 Find the Surface Area of a Prism
The Skyscraper is 414

meters high. The base is a square with sides that are 64 meters. What is the surface area of the skyscraper?

414

64(64)=4096

64(414)=26496

Surface Area = 4(64•414)+2(64•64)=114,176 m2

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Example 1 Find the Surface Area of a Prism
The Skyscraper is 414

meters high. The base is a square with sides that are 64 meters. What is the surface area of the skyscraper?

414

Surface Area = 4(64•414)+2(64•64)=114,176 m2

Surface Area = (4•64)•414+2(64•64)=114,176 m2

Perimeter of the base

Area of the base

height

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Nets
A net is a pattern that can be cut and folded

to form a polyhedron.

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Surface Area of a Right Prism
The surface area, S, of a

right prism is S = 2B + Ph where B is the area of a base, P is the perimeter of a base, and h is the height

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Example 2 Finding the Surface Area of a Prism
Find the surface area

of each right prism

5 in.

12 in.

8 in.

5 in.

12 in.

4 in.

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Example 2 Finding the Surface Area of a Prism
Find the surface area

of each right prism

S = 2B + Ph
S = 2(60) +(34)•8
S = 120 + 272 = 392 in2

Area of the Base = 5x12=60

Perimeter of Base = 5+12+5+12 = 34

S = 2B + Ph

Height of Prism = 8

5 in.

12 in.

8 in.

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Example 2 Finding the Surface Area of a Prism
Find the surface area

of each right prism

Area of the Base = ½(5)(12)=30

Perimeter of Base = 5+12+13=30

S = 2B + Ph

Height of Prism = 4 (distance between triangles)

S = 2B + Ph
S = 2(30) +(30)•4
S = 60 + 120 = 180 in2

5 in.

12 in.

4 in.

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Cylinders
A cylinder is a solid with congruent circular bases that lie

in parallel planes
The altitude, or height, of a cylinder is the perpendicular distance between its bases
The lateral area of a cylinder is the area of its curved lateral surface.
A cylinder is right if the segment joining the centers of its bases is perpendicular to its bases

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Surface Area of a Right Cylinder
The surface area, S, of a

right circular cylinder is S = 2B + Ch or 2πr2 + 2πrh where B is the area of a base, C is the circumference of a base, r is the radius of a base, and h is the height

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Example 3 Finding the Surface Area of a Cylinder
Find the surface area

of the cylinder

3 ft

4 ft

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Example 3 Finding the Surface Area of a Cylinder
Find the surface area

of the cylinder
2πr2 + 2πrh
2π(3)2 + 2π(3)(4)
18π + 24π
42π ≈ 131.9 ft2

3 ft

4 ft

Radius = 3
Height = 4

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Pyramids
A pyramid is a polyhedron in which the base is a

polygon and the lateral faces are triangles that have a common vertex

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Pyramids
The intersection of two lateral faces is a lateral edge
The

intersection of the base and a lateral face is a base edge
The altitude or height of the pyramid is the perpendicular distance between the base and the vertex

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Regular Pyramid
A pyramid is regular if its base is a regular

polygon and if the segment from the vertex to the center of the base is perpendicular to the base
The slant height of a regular pyramid is the altitude of any lateral face (a nonregular pyramid has no slant height)

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Developing the formula for surface area of a regular pyramid
Area of

each triangle is ½bL
Perimeter of the base is 6b
Surface Area = (Area of base) + 6(Area of lateral faces)
S = B + 6(½bl)
S = B + ½(6b)(l)
S= B + ½Pl

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Surface Area of a Regular Pyramid
The surface area, S, of a

regular pyramid is S = B + ½Pl Where B is the area of the base, P is the perimeter of the base, and L is the slant height

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Example 1 Finding the Surface Area of a Pyramid
Find the surface area

of each regular pyramid

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Example 1 Finding the Surface Area of a Pyramid
Find the surface area

of each regular pyramid

Base is a Square Area of Base = 5(5) = 25

Perimeter of Base 5+5+5+5 = 20

Slant Height = 4

S = B + ½PL

S = 25 + ½(20)(4) = 25 + 40 = 65 ft2

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Example 1 Finding the Surface Area of a Pyramid
Find the surface area

of each regular pyramid

S = B + ½PL

Base is a Hexagon A=½aP

Perimeter = 6(6)=36

Slant Height = 8

S = + ½(36)(8) = + 144 = 237.5 m2

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Cones
A cone is a solid that has a circular base and

a vertex that is not in the same plane as the base
The lateral surface consists of all segments that connect the vertex with point on the edge of the base
The altitude, or height, of a cone is the perpendicular distance between the vertex and the plane that contains the base

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Right Cone
A right cone is one in which the vertex lies

directly above the center of the base
The slant height of a right cone is the distance between the vertex and a point on the edge of the base

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Developing the formula for the surface area of a right cone
Use

the formula for surface area of a pyramid S = B + ½Pl
As the number of sides on the base increase it becomes nearly circular
Replace ½P (half the perimeter of the pyramids base) with πr (half the circumference of the cone's base)

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Surface Area of a Right Cone
The surface area, S, of a

right cone is S = πr2 + πrl Where r is the radius of the base and L is the slant height of the cone

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Example 2 Finding the Surface Area of a Right Cone
Find the surface

area of the right cone

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Example 2 Finding the Surface Area of a Right Cone
Find the surface

area of the right cone

S = πr2 + πrl
= π(5)2 + π(5)(7)
= 25π + 35π
= 60π or 188.5 in2

Radius = 5
Slant height = 7

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Volume formulas
The Volume, V, of a prism is V = Bh
The

Volume, V, of a cylinder is V = πr2h
The Volume, V, of a pyramid is V = 1/3Bh
The Volume, V, of a cone is V = 1/3πr2h
The Surface Area, S, of a sphere is S = 4πr2
The Volume, V, of a sphere is V = 4/3πr3

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Volume
The volume of a polyhedron is the number of cubic units

contained in its interior
Label volumes in cubic units like cm3, in3, ft3, etc

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Postulates
All the formulas for the volumes of polyhedrons are based on

the following three postulates
Volume of Cube Postulate: The volume of a cube is the cube of the length of its side, or V = s3
Volume Congruence Postulate: If two polyhedrons are congruent, then they have the same volume
Volume Addition Postulate: The volume of a solid is the sum of the volumes of all its nonoverlapping parts

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Example 1: Finding the Volume of a Rectangular Prism
The cardboard box

is 5" x 3" x 4" How many unit cubes can be packed into the box? What is the volume of the box?

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Example 1: Finding the Volume of a rectangular Prism
The cardboard box

is 5" x 3" x 4" How many unit cubes can be packed into the box? What is the volume of the box?

How many cubes in bottom layer?
5(3) = 15
How many layers?
4
V=5(3)(4) = 60 in3

V = L x W x H for a rectangular prism

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Volume of a Prism and a Cylinder
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

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Volume of a Prism
The Volume, V, of a prism is V =

Bh where B is the area of a base and h is the height

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Volume of a Cylinder
The volume, V, of a cylinder is V=Bh

or V = πr2h where B is the area of a base, h is the height and r is the radius of a base

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Example 2 Finding Volumes
Find the volume of the right prism and the

right cylinder

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Example 2 Finding Volumes
Find the volume of the right prism and the

right cylinder

3 cm

4 cm

Area of Base
B = ½(3)(4)=6
Height = 2

V = Bh
V = 6(2)
V = 12 cm3

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Example 2 Finding Volumes
Find the volume of the right prism and the

right cylinder

Area of Base
B = π(7)2 =49π
Height = 5

V = Bh
V = 49π(5)
V = 245π in3

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Example 3 Estimating the Cost of Moving
You are moving from Newark, New

Jersey, to Golden, Colorado - a trip of 2000 miles. Your furniture and other belongings will fill half the truck trailer. The moving company estimates that your belongings weigh an average of 6.5 pounds per cubic foot. The company charges $600 to ship 1000 pounds. Estimate the cost of shipping your belongings.