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10.5 NOTES
Objectives
 Learn and apply the formula for the surface area of a pyramid.
 Learn and apply the formula for the surface area of a cone.
Vocabulary
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vertex of a pyramid
regular pyramid
slant height of a regular pyramid
altitude of a pyramid
vertex of a cone
axis of a cone
right cone
oblique cone
slant height of a right cone
altitude of a cone
1. The
is the point opposite the base of the pyramid.
2. The base of a
the lateral faces
are congruent isosceles triangles.
3. The
midpoint of an edge of the base.
4. The
vertex to the plane of the base.
is a regular polygon, and
is the distance from the vertex to the
is the perpendicular segment from the
1
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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Example 1: Finding Lateral Area and Surface Area of Pyramids
Find the lateral area and surface area of a regular square pyramid with base
edge length 14 cm and slant height 25 cm. Round to the nearest tenth, if
necessary.
Example 2: Finding Lateral Area and Surface Area of Pyramids
Find the lateral area and surface area of the regular pyramid.
Step 1 Find the base perimeter and apothem.
The base perimeter is 6(10) = 60 in.
The apothem is 5 3 in , so the base area is
1
1
aP  (5 3)(60)  150 3 in 2
2
2
Step 2 Find the lateral area.
1
L  Pl
Lateral area of a regular pyramid
2
1
=  60 16   480 in 2 Substitute 60 for P and 16 for l
2
3
Step 3 Find the surface area.
1
Pl + B
Surface area of a regular pyramid
2
=480  150 3  739.8 in 2 Substitute 150 3 for B.
S
Example 3
Find the lateral area and surface area of a regular triangular pyramid with
base edge length 6 ft and slant height 10 ft.
Step 1 Find the base perimeter and apothem.
Step 2 Find the lateral area.
Step 3 Find the surface area.
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5. The
is the distance from the vertex of a right
cone to a point on the edge of the base.
6, The
is a perpendicular segment from the vertex
of the cone to the plane of the base.
Example 4: Finding Lateral Area and Surface Area of Right Cones
Find the lateral area and surface area of a right cone with radius 9 cm and
slant height 5 cm.
Example 5: Finding Lateral Area and Surface Area of Right Cones
Find the lateral area and surface area of the cone.
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Example 6
Find the lateral area and surface area of the right cone.
Example 7: Exploring Effects of Changing Dimensions
The base edge length and slant height of the regular hexagonal pyramid are
both divided by 5. Describe the effect on the surface area.
3 in.
2 in.
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Example 8
The base edge length and slant height of the regular square pyramid are both
multiplied by
2
. Describe the effect on the surface area.
3
8 ft
10 ft
Example 9: Finding Surface Area of Composite Three-Dimensional
Figures
Find the surface area of the composite figure.
Left-hand cone:
Right-hand cone:
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Composite figure
Example 10
Find the surface area of the composite figure.
Surface Area of Cube without the top side:
Surface Area of Pyramid without base:
Surface Area of Composite:
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Example 11: Manufacturing Application
If the pattern shown is used to make a paper cup, what is the diameter
of the cup?
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
1
of the pattern is also
of the area of the large circle, so
2
Example 12
What if…? If the radius of the large circle were 12 in., what would be
the radius of the cone?
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