11.3 Notes.notebook
April 07, 2013
You are building a model of a clock tower. You have already constructed the basic structure of the tower at the right. Now you want to paint the roof. How much area does the paint need to cover? Give your answer in square inches. Explain your method.
156.2 in
2
Apr 5­3:13 PM
1
11.3 Notes.notebook
April 07, 2013
11.3 Surface Area of Pyramids and Cones
V
E
h
Pentagonal pyramid
V­ABCDE
D
A
B
C
Jun 3­7:45 PM
2
11.3 Notes.notebook
April 07, 2013
V
E
h
Pentagonal pyramid
V­ABCDE
Point V is the vertex of the pyramid
and pentagon ABCDE is the base.
D
A
The segment from the vertex perpendicular to the base is the altitude and its length is the height, h, of the pyramid.
B
C
Jun 3­7:45 PM
3
11.3 Notes.notebook
April 07, 2013
V
The five triangular faces with V in common, such as ΔVAB, are lateral faces.
E
h
These faces intersect in segments called lateral edges.
D
A
B
C
Jun 3­7:45 PM
4
11.3 Notes.notebook
April 07, 2013
The height of a lateral face is called the slant height of the pyramid. It is denoted by l .
h
O
l
Jun 3­7:45 PM
5
11.3 Notes.notebook
April 07, 2013
Most of the pyramids you will study will be regular pyramids.
These are pyramids with the following properties:
(1) The base is a regular polygon
(2) All lateral edges are congruent
(3) All lateral faces are congruent isosceles triangles.
(4) The altitude meets the base at its center.
Jun 3­7:45 PM
6
11.3 Notes.notebook
April 07, 2013
Apr 5­3:14 PM
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11.3 Notes.notebook
April 07, 2013
V
A regular square pyramid has base edges 10 and lateral edges 13. Find:
• the slant height
• height
13
h
l
C
D
M
O
A
10
B
Jun 3­8:07 PM
8
11.3 Notes.notebook
April 07, 2013
V
A regular square pyramid has base edges 10 and lateral edges 13. Find:
• the slant height
• height
13
12
h
C
D
5
M
5
O
A
10
5
B
Jun 3­8:07 PM
9
11.3 Notes.notebook
April 07, 2013
V
A regular square pyramid has base edges 10 and lateral edges 13. Find:
• the lateral area
13
l
h
C
D
5
M
5
O
A
5
B
10
V
13
13
B
10
C
Jun 3­8:07 PM
10
11.3 Notes.notebook
April 07, 2013
Theorem
The lateral area of a regular pyramid equals half the perimeter of the base times the slant height.
L.A. = ½pl
h
O
l
b
Jun 3­8:07 PM
11
11.3 Notes.notebook
L.A. = ( ½bl ) n
= ½( bn ) l
April 07, 2013
h
Since bn = p
O
L.A. = ½pl
l
b
Jun 3­8:07 PM
12
11.3 Notes.notebook
April 07, 2013
regular pyramid with To find the lateral area of a • Find the area of one lateral face and multiply by OR
• Use the formula L.A. = n lateral faces:
n
½pl
V
V
E
h
l
D
A
h
13
h
C
D
5
M
5
B
O
C
A
10
O
l
5
B
Jun 3­8:07 PM
13
11.3 Notes.notebook
April 07, 2013
To find the surface area (or total area) of a pyramid, add the area of its base to the lateral area.
T.A. = L.A. + B
V
V
E
h
l
D
A
h
13
h
C
D
5
M
5
B
O
C
A
10
O
l
5
B
Apr 5­3:14 PM
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11.3 Notes.notebook
April 07, 2013
What is the surface area of the hexagonal pyramid?
256 in
2
Apr 5­3:16 PM
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11.3 Notes.notebook
April 07, 2013
A square pyramid has base edges of 5 m and a slant height of 3 m. What is the surface area of the pyramid?
55 m
2
Apr 5­3:16 PM
16
11.3 Notes.notebook
April 07, 2013
Suppose the slant height of a pyramid is doubled. How does this affect the lateral area of the pyramid? Explain.
The L.A. will double. Since L.A. = ½pl, then replacing l with 2l
L.A = ½p(2l)
= 2 (½pl)
= 2(L.A.)
Apr 5­3:16 PM
17
11.3 Notes.notebook
April 07, 2013
What is the surface area of the square pyramid with base edges of 8 cm and a slant height of 12 cm?
256 cm
2
Apr 5­3:29 PM
18
11.3 Notes.notebook
April 07, 2013
The Pyramid of Cestius is located in Rome, Italy. Using the dimensions in the figure below, what is the lateral area of the Pyramid of Cestius? Round to the nearest square meter.
Apr 5­3:17 PM
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11.3 Notes.notebook
April 07, 2013
Apr 5­3:19 PM
20
11.3 Notes.notebook
April 07, 2013
What is the lateral are of the hexagonal pyramid? Round to the nearest square foot.
5649 ft
2
Apr 5­3:20 PM
21
11.3 Notes.notebook
April 07, 2013
How does the slant height of a regular pyramid relate to its height? Explain.
The slant height is the hypotenuse of a right triangle with a leg length equal to the height of the pyramid, so the slant height is greater than the height.
Apr 5­3:20 PM
22
11.3 Notes.notebook
April 07, 2013
What is the lateral area of a pyramid with a height of 8 in. and a square base that measure 9 in. on each side? Round to the nearest tenth.
165.2 in
2
Apr 5­3:30 PM
23
11.3 Notes.notebook
April 07, 2013
What is the lateral area of the square pyramid?
60 m
2
Apr 5­3:24 PM
24
11.3 Notes.notebook
What is the 85 m
April 07, 2013
surface area of the square pyramid?
2
Apr 5­3:24 PM
25
11.3 Notes.notebook
April 07, 2013
V
10
A regular triangular pyramid has lateral edges 10 and height 6. Find the lateral area.
6
l
B
A
O
M
Recall the centroid.
The medians of a triangle intersect in a 2 / of the distance from each point that is 3
vertex to the midpoint of the opposite side.
C
AO = 2 / 3 AM
Jun 3­8:07 PM
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11.3 Notes.notebook
April 07, 2013
, ight
t he
slan
l
h
, ight
t he
slan
l
A cone is like a pyramid except that its bases are circles instead of polygons.
Jun 4­4:09 PM
27
11.3 Notes.notebook
April 07, 2013
A cone is like a pyramid except that its bases are circles instead of polygons.
These are oblique cones.
Slant height only applies to a regular pyramid and a right cone.
Henceforth the word "cone" will always refer to a right cone.
Jun 4­4:09 PM
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11.3 Notes.notebook
April 07, 2013
Apr 5­3:20 PM
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11.3 Notes.notebook
April 07, 2013
Recall in a pyramid,
L.A. = 1 / 2 ph
Thus....
Jun 4­4:09 PM
30
11.3 Notes.notebook
April 07, 2013
, ight
t he
slan
l
h
Theorem
The lateral area of a cone equals half the circumference of the base times the slant height.
r
L.A. = 1 / 2 (2 πr ) l
L.A. = πrl
Jun 4­4:09 PM
31
11.3 Notes.notebook
April 07, 2013
To find the surface area (or total area) of a cone, add the area of base to the lateral area.
its T.A. = L.A. + B
, ight
t he
slan
l
h
r
Apr 7­11:39 AM
32
11.3 Notes.notebook
Find the the cone.
lateral area April 07, 2013
and the total area
of l
6
3
Jun 4­4:27 PM
33
11.3 Notes.notebook
Find the lateral area and the cone.
April 07, 2013
the total area
of l
6
3
Jun 4­4:27 PM
34
11.3 Notes.notebook
April 07, 2013
What is the surface area of the cone in terms of π?
600π cm2
Apr 5­3:21 PM
35
11.3 Notes.notebook
April 07, 2013
The radius of the base of a cone is 16 m. Its slant height is 28 m. What is the surface area in terms of π?
704 π m2
Apr 5­3:22 PM
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11.3 Notes.notebook
April 07, 2013
What is the lateral area of the cone in terms of π?
65 π cm2
Apr 5­3:30 PM
37
11.3 Notes.notebook
April 07, 2013
By cutting a cone and laying it out flat, you can see how the formula for the lateral area of a cone L.A. = ½(C base )(l)
resembles that for the area of a triangle
A. = ½(b)(h)
Apr 5­3:22 PM
38
11.3 Notes.notebook
April 07, 2013
In a chemistry lab experiment, you use the conical filter funnel shown. How much filter paper do you need to line the funnel?
7566 mm
2
Apr 5­3:22 PM
39
11.3 Notes.notebook
April 07, 2013
What is the lateral area of a traffic cone with radius 10 in. and height 28 in.? Round to the nearest whole number.
934 in.
2
Apr 5­3:23 PM
40
11.3 Notes.notebook
April 07, 2013
Suppose the radius of a cone is halved, but the slant height remains the same. How does this affect the lateral area of the cone? Explain.
The L.A. will be halved.
Since L.A. = πrl, then replacing r with r
/2
= π( r / 2 ) l
= ½(π rl )
= ½(L.A.)
Apr 5­3:23 PM
41
11.3 Notes.notebook
April 07, 2013
What is the lateral are of the ice cream cone shown? Round to the nearest square centimeter.
about 113 cm
2
Apr 5­3:31 PM
42
11.3 Notes.notebook
April 07, 2013
What is the lateral area of the cone?
2π√29 ft 2 , or about 33.8 ft
2
Apr 5­3:24 PM
43
11.3 Notes.notebook
April 07, 2013
What is the surface area of the cone?
(2π√29 + 4π) ft 2 , or about 46.4 ft
2
Apr 5­3:24 PM
44
11.3 Notes.notebook
April 07, 2013
How do the height and the slant height of a pyramid differ?
The height is the distance from the vertex to the center of the base, while the slant height is the distance from the vertex to the midpoint of an edge of the base.
Apr 5­3:24 PM
45