Geometry Surface Area of Pyramids and Cones

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Geometry Surface Area of Pyramids and Cones
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Warm Up
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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.
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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.
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Lateral and surface Area of a regular 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.
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Finding Lateral and surface Area of Pyramids
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b.)Find the lateral area and surface area of each
regular pyramid.
Step 1
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 2
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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.
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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.
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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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Lateral and Surface Area of a right cone
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Finding Lateral Area and Surface Area of right
cones
A) A right cone with radius 2 m and slant height
3 m.
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B)
Step 2
Find the lateral area and surface area.
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Now you try!
2) Find the lateral area and surface area of the
right cone.
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Exploring Effects of Changing Dimensions
The radius and slant height of the right cone
tripled. Describe the effect on the surface area.
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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.
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Finding Surface Area of Composite
Three-Dimensional Figures
Find the surface area of the composite figure.
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Now you try!
4) Find the surface area of the composite figure.
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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?
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Now you try!
5) If the radius of the large circle were 12 in.,
what would be the radius of the cone?
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Now some problems for you to practice !
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Assessment
1) Describe the endpoints of an axis of a cone.
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2) Find the lateral area and surface area of each
regular pyramid.
b.
a.
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b.
a.
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4) Describe the effect of each change on the
surface area of the given figure.
b. The dimension are tripled.
a. The dimensions are cut in half.
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5) Find the surface area of each composite figure.
b.
a.
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6) Anna is making a birthday hat from a pattern
that is ¾ of a circle of colored paper. If Annas
head is 7 inches in diameter, will the hat fit
her? Explain.
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Lets review
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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.
Next page
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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.
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Lateral and surface Area of a regular 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.
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Finding Lateral and surface Area of Pyramids
Next page
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b.)Find the lateral area and surface area of each
regular pyramid.
Step 1
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 2
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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.
Next page
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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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Lateral and Surface Area of a right cone
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Finding Lateral Area and Surface Area of right
cones
A) A right cone with radius 2 m and slant height
3 m.
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B)
Step 2
Find the lateral area and surface area.
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Exploring Effects of Changing Dimensions
The radius and slant height of the right cone
tripled. Describe the effect on the surface area.
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Finding Surface Area of Composite
Three-Dimensional Figures
Find the surface area of the composite figure.
Next page
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(No Transcript)
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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?
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You did a great job today!