confidential 1 geometry surface area of pyramids and cones
TRANSCRIPT
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CONFIDENTIAL 1
GeometryGeometry
Surface Area of Surface Area of Pyramids and Cones Pyramids and Cones
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CONFIDENTIAL 2
Warm UpWarm Up
15 in.
10 in.
17 in.
10 cm
15 cm
8 cm
Find the surface area of each right prism or right cylinder. Round your answer to the nearest tenth.
2 cm
3 cm
1. 2. 3.
1) 490 in2 2) 62.83 cm2 3) 700 cm2
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CONFIDENTIAL 3
Surface Area of Pyramids and Cones Surface Area of Pyramids and Cones
The vertex of a pyramid is the point opposite the base of the pyramid. The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base.
regular pyramid nonregular pyramid
Slant height
Vertices
Lateral faces
Altitude
Bases
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CONFIDENTIAL 4
The lateral faces of a regular pyramid can be arranged to cover half of a rectangle with a height equal to the slant height of the pyramid. The width of the rectangle is equal to the base perimeter of the pyramid.
s
s s
sl
P = 4s
l
ssss
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CONFIDENTIAL 5
Lateral and surface Area of a Lateral and surface Area of a regular pyramidregular pyramid
The lateral area of a regular pyramid with perimeter P and slant height l is L = 1/2Pl.
The surface area of a regular pyramid with lateral area L and base area B is S = L + B, or S = ½ pl + B.
l
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CONFIDENTIAL 6
Finding Lateral and surface Area of Pyramids
Find the lateral area and surface area of each pyramid.a.) a regular square pyramid with base edge length 5 in. and slant height 9in.
L = ½ Pl Lateral area of a regular pyramid = ½(20)(9) = 90 in P = 4(5) = 20 in.S = ½ Pl + B Surface area of a regular pyramid = 90 + 25 = 115 in B = 5 = 25 in
2
2 2 2
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CONFIDENTIAL 7
b.)Find the lateral area and surface area of each regular pyramid.
7 m
4 m
find the base perimeter and apothem.The base perimeter is 6(4) = 24 m.The apothem is 2 3 m, so the base area is ½ aP = ½ (2 3) (24) = 24 3 m 2
Step 1
Step 2 Find the lateral area.L = ½ Pl Lateral area of a regular pyramid = ½ (24)(7) = 84 m Substitute 24 for P and 7 for l.
Step 3 Find the surface area.S = 1/2 Pl + B Surface area of a regular pyramid = 84 + 24 3 = 125.6 cm Substitute 24 3 for B. 2
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CONFIDENTIAL 8
Now you try!
1) Find the lateral area and surface area of a regular triangular pyramid with base edge length 6 ft and slant height 10ft.
1) lateral area = 90 ft2 surface area = 105 ft2
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CONFIDENTIAL 9
The vertex of a cone is the point opposite the base. The axis of a cone is the segment with endpoints at the vertex and the center of the base. The axis of a right cone is perpendicular to the base. The axis of an oblique cone is not perpendicular to the base.
Slant height
right cone oblique cone
Vertices
Lateral surfaces
axis axis
Base
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CONFIDENTIAL 10
The slant height of a right cone is the distance from the vertex of a right cone to a point on the edge of the base. The altitude of a cone is a perpendicular segment from the vertex of the cone to the plane of the base.
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CONFIDENTIAL 11
Lateral and Surface Area of a right cone
Lateral and Surface Area of a right cone
r
l
rl
The lateral area of a right cone with radius r and slant height l is L = rl.
The surface area of a right cone with lateral area L and base area B is S = L + B, or S = rl + r 2
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CONFIDENTIAL 12
Finding Lateral Area and Surface Area of right cones
Find the lateral area and surface area of each cone. Give your answers in terms of .
A) A right cone with radius 2 m and slant height 3 m.
L = rl Lateral area of a cone = (2) (3) = 6 m2 Substitute 2 for r and 3 for l.S = rl + r2 Surface area of a cone = 6 + (2)2 = 10 m2 Substitute 2 for r and 3 for l.
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CONFIDENTIAL 13
B)
5 ft
l12 ft
Step 1 Use the Pythagorean Theorem to find l.
l = 5 + 12 = 13 ft2 2
Step 2 Find the lateral area and surface area.
L = rl Lateral area of a right cone = (5) (13) = 65 ft2 Substitute 5 for r and 13 for l.S = rl + r2 Surface area of a right cone = 65 + (5)2 = 90 ft2 Substitute 5 for r and 13 for l.
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CONFIDENTIAL 14
Now you try!
16 ft
6 ft
2) Find the lateral area and surface area of the right cone.
2) lateral area = 251.43 ft2 surface area = 452.57 ft2
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CONFIDENTIAL 15
Exploring Effects of Changing Dimensions
3 cm
5 cmThe radius and slant height of the right cone tripled. Describe the effect on the surface area.
Notice that 216 = 9(24). I f the radius and slant heightare tripled, the surface area is multiplied by 32, or 9.
S = rl + r2 = (9)(15) + (9)2 = 216 cm2
S = rl + r2 = (3)(5) + (3)2 = 24 cm2
Original dimensions: Radius and slant height tripled:
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CONFIDENTIAL 16
Now you try!
3) The base edge length and slant height of the regular square pyramid are both multiplied by 2/3 . Describe the effect on the surface area.
12 ft
15 ft
3) The surface area is multiplied by 4/9.
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CONFIDENTIAL 17
Finding Surface Area of Composite Three-Dimensional Figures
Finding Surface Area of Composite Three-Dimensional Figures
Find the surface area of the composite figure.
28 cm
90
cm
45
cm
The height of the cone is 90 - 45 = 45 cm.By the Pythagorean Theorem,l = 282 + 452 = 53 cm. the lateral area of thecoan isL = rl = (28)(53) = 1484 cm2.
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CONFIDENTIAL 18
28 cm
90
cm
45
cmThe lateral area of the cylinder is L = 2rh = 2(28)(45) =2520 cm2.
The base area is B = r2 = (28)2 = 784 cm2
S = (cone lateral area) + (cylinder lateral area) + (base area) = 2520 + 784 + 1484 = 4788 cm2
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CONFIDENTIAL 19
Now you try!
4) Find the surface area of the composite figure.
2 yd
2 yd
2 yd4) 20.94 yd2
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CONFIDENTIAL 20
Electronics Application
Tim is replacing the paper cone of an antique speaker. He measured the existing cone and created the pattern for the lateral
surface from a large circle. What is the diameter of the cone?
10 in.
The radius of the large circle used to create the pattern is the slant height of the cone.
The area of the pattern is the lateral area of the cone. The area of the pattern is also ¾ of the area of the large circle, so rl = ¾ r . r(10) = ¾ (10) = 7.5 in.The diameter of the cone is 2(7.5) = 15 in.
2
2Substitute 10 for l, the slant height of the cone and the radius of the large circle.Solve it
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CONFIDENTIAL 21
Now you try!
5) If the radius of the large circle were 12 in., what would be the radius of the cone?
5) 9 in.
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CONFIDENTIAL 22
Now some problems for you to practice !
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CONFIDENTIAL 23
Assessment
1) Describe the endpoints of an axis of a cone.
1) The endpoints at the vertex and the center of the base of the cone joins to form the axis of the cone.
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CONFIDENTIAL 24
2) Find the lateral area and surface area of each regular pyramid.
a. b.
8 cm
12 cm15 ft
16 ft
16 ft
2b) lateral area = 480 ft2 surface area = 1472 ft2
2a) lateral area = 288 cm2 surface area = 576 cm2
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CONFIDENTIAL 25
3) Find the lateral area and surface of each right cone. Given your answer in terms of .
24 in.
25 in.
14 m.
22 m.
a. b.
3b) lateral area = 308∏ m2 surface area = 504∏ m2
3a) lateral area = 175∏ in2 surface area = 224∏ in2
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CONFIDENTIAL 26
4) Describe the effect of each change on the surface area of the given figure.
a. The dimensions are cut in half.
b. The dimension are tripled.
9 cm.
15 cm.10 in.
6 in.
6 in.
4b) lateral area = 308∏ m2 surface area = 504∏ m2
4a) lateral area = 175∏ in2 surface area = 224∏ in2
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CONFIDENTIAL 27
5) Find the surface area of each composite figure.
15 m.
12 m.
32 m.
26 m.15 ft.
8 ft.
18 ft.
a. b.
5b) 3498.95 m2 5a) 1184 ft2
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CONFIDENTIAL 28
6) Anna is making a birthday hat from a pattern that is ¾ of a circle of colored paper. If Anna’s head is 7 inches in diameter, will the hat fit her? Explain.
6 in.
6 ) Anna’s head will fit in the hat as the diameter of the cone will be 9 inches.
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CONFIDENTIAL 29
Let’s review
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CONFIDENTIAL 30
Surface Area of Pyramids and Cones Surface Area of Pyramids and Cones
The vertex of a pyramid is the point opposite the base of the pyramid. The base of a regular pyramid is a regular polygon, and the lateral faces are congruent isosceles triangles. The slant height of a regular pyramid is the distance from the vertex to the midpoint of an edge of the base. The altitude of a pyramid is the perpendicular segment from the vertex to the plane of the base.
regular pyramid nonregular pyramid
Slant height
Vertices
Lateral faces
Altitude
Bases
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CONFIDENTIAL 31
The lateral faces of a regular pyramid can be arranged to cover half of a rectangle with a height equal to the slant height of the pyramid. The width of the rectangle is equal to the base perimeter of the pyramid.
s
s s
sl
P = 4s
l
ssss
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CONFIDENTIAL 32
Lateral and surface Area of a Lateral and surface Area of a regular pyramidregular pyramid
The lateral area of a regular pyramid with perimeter P and slant height l is L = 1/2Pl.
The surface area of a regular pyramid with lateral area L and base area B is S = L + B, or S = ½ pl + B.
l
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CONFIDENTIAL 33
Finding Lateral and surface Area of Pyramids
Find the lateral area and surface area of each pyramid.a.) a regular square pyramid with base edge length 5 in. and slant height 9in.
L = ½ Pl Lateral area of a regular pyramid = ½(20)(9) = 90 in P = 4(5) = 20 in.S = ½ Pl + B Surface area of a regular pyramid = 90 + 25 = 115 in B = 5 = 25 in
2
2 2 2
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CONFIDENTIAL 34
b.)Find the lateral area and surface area of each regular pyramid.
7 m
4 m
find the base perimeter and apothem.The base perimeter is 6(4) = 24 m.The apothem is 2 3 m, so the base area is ½ aP = ½ (2 3) (24) = 24 3 m 2
Step 1
Step 2 Find the lateral area.L = ½ Pl Lateral area of a regular pyramid = ½ (24)(7) = 84 m Substitute 24 for P and 7 for l.
Step 3 Find the surface area.S = 1/2 Pl + B Surface area of a regular pyramid = 84 + 24 3 = 125.6 cm Substitute 24 3 for B. 2
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CONFIDENTIAL 35
The vertex of a cone is the point opposite the base. The axis of a cone is the segment with endpoints at the vertex and the center of the base. The axis of a right cone is perpendicular to the base. The axis of an oblique cone is not perpendicular to the base.
Slant height
right cone oblique cone
Vertices
Lateral surfaces
axis axis
Base
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CONFIDENTIAL 36
The slant height of a right cone is the distance from the vertex of a right cone to a point on the edge of the base. The altitude of a cone is a perpendicular segment from the vertex of the cone to the plane of the base.
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CONFIDENTIAL 37
Lateral and Surface Area of a right cone
Lateral and Surface Area of a right cone
r
l
rl
The lateral area of a right cone with radius r and slant height l is L = rl.
The surface area of a right cone with lateral area L and base area B is S = L + B, or S = rl + r 2
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CONFIDENTIAL 38
Finding Lateral Area and Surface Area of right cones
Find the lateral area and surface area of each cone. Give your answers in terms of .
A) A right cone with radius 2 m and slant height 3 m.
L = rl Lateral area of a cone = (2) (3) = 6 m2 Substitute 2 for r and 3 for l.S = rl + r2 Surface area of a cone = 6 + (2)2 = 10 m2 Substitute 2 for r and 3 for l.
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CONFIDENTIAL 39
B)
5 ft
l12 ft
Step 1 Use the Pythagorean Theorem to find l.
l = 5 + 12 = 13 ft2 2
Step 2 Find the lateral area and surface area.
L = rl Lateral area of a right cone = (5) (13) = 65 ft2 Substitute 5 for r and 13 for l.S = rl + r2 Surface area of a right cone = 65 + (5)2 = 90 ft2 Substitute 5 for r and 13 for l.
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CONFIDENTIAL 40
Exploring Effects of Changing Dimensions
3 cm
5 cmThe radius and slant height of the right cone tripled. Describe the effect on the surface area.
Notice that 216 = 9(24). I f the radius and slant heightare tripled, the surface area is multiplied by 32, or 9.
S = rl + r2 = (9)(15) + (9)2 = 216 cm2
S = rl + r2 = (3)(5) + (3)2 = 24 cm2
Original dimensions: Radius and slant height tripled:
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CONFIDENTIAL 41
Finding Surface Area of Composite Three-Dimensional Figures
Finding Surface Area of Composite Three-Dimensional Figures
Find the surface area of the composite figure.
The height of the cone is 90 - 45 = 45 cm.By the Pythagorean Theorem,l = 282 + 452 = 53 cm. the lateral area of thecoan isL = rl = (28)(53) = 1484 cm2.
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28 cm
90
cm
45
cm
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CONFIDENTIAL 42
28 cm
90
cm
45
cmThe lateral area of the cylinder is L = 2rh = 2(28)(45) =2520 cm2.
The base area is B = r2 = (28)2 = 784 cm2
S = (cone lateral area) + (cylinder lateral area) + (base area) = 2520 + 784 + 1484 = 4788 cm2
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CONFIDENTIAL 43
Electronics Application
Tim is replacing the paper cone of an antique speaker. He measured the existing cone and created the pattern for the lateral surface from a large circle. What is the diameter of the cone?
10 in.
The radius of the large circle used to create the pattern is the slant height of the cone.
The area of the pattern is the lateral area of the cone. The area of the pattern is also ¾ of the area of the large circle, so rl = ¾ r . r(10) = ¾ (10) = 7.5 in.The diameter of the cone is 2(7.5) = 15 in.
2
2Substitute 10 for l, the slant height of the cone and the radius of the large circle.Solve it
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CONFIDENTIAL 44
You did a You did a greatgreat job job today!today!