Surface Area of

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Surface Area of Prisms and Cylinders
10-4
Holt Geometry
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Warm Up Find the perimeter and area of each
polygon. 1. a rectangle with base 14 cm and
height 9 cm 2. a right triangle with 9 cm and 12
cm legs 3. an equilateral triangle with side
length 6 cm
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Objectives
Learn and apply the formula for the surface area
of a prism. Learn and apply the formula for the
surface area of a cylinder.
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Vocabulary
lateral face lateral edge right prism oblique
prism altitude surface area lateral surface axis
of a cylinder right cylinder oblique cylinder
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Prisms and cylinders have 2 congruent parallel
bases. A lateral face is not a base. The edges of
the base are called base edges. A lateral edge is
not an edge of a base. The lateral faces of a
right prism are all rectangles. An oblique prism
has at least one nonrectangular lateral face.
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An altitude of a prism or cylinder is a
perpendicular segment joining the planes of the
bases. The height of a three-dimensional figure
is the length of an altitude.
Surface area is the total area of all faces and
curved surfaces of a three-dimensional figure.
The lateral area of a prism is the sum of the
areas of the lateral faces.
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The net of a right prism can be drawn so that the
lateral faces form a rectangle with the same
height as the prism. The base of the rectangle is
equal to the perimeter of the base of the prism.
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The surface area of a right rectangular prism
with length l, width w, and height h can be
written as S 2lw 2wh 2lh.
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Example 1A Finding Lateral Areas and Surface
Areas of Prisms
Find the lateral area and surface area of the
right rectangular prism. Round to the nearest
tenth, if necessary.
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Example 1B Finding Lateral Areas and Surface
Areas of Prisms
Find the lateral area and surface area of a right
regular triangular prism with height 20 cm and
base edges of length 10 cm. Round to the nearest
tenth, if necessary.
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The lateral surface of a cylinder is the curved
surface that connects the two bases. The axis of
a cylinder is the segment with endpoints at the
centers of the bases. The axis of a right
cylinder is perpendicular to its bases. The axis
of an oblique cylinder is not perpendicular to
its bases. The altitude of a right cylinder is
the same length as the axis.
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Example 2A Finding Lateral Areas and Surface
Areas of Right Cylinders
Find the lateral area and surface area of the
right cylinder. Give your answers in terms of ?.
The radius is half the diameter, or 8 in.
L 2?rh 2?(8)(10) 160? in2
S L 2?r2 160? 2?(8)2 288? in2
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Example 2B Finding Lateral Areas and Surface
Areas of Right Cylinders
Find the lateral area and surface area of a right
cylinder with circumference 24? cm and a height
equal to half the radius. Give your answers in
terms of ?.
Step 1 Use the circumference to find the radius.
Circumference of a circle
C 2?r
24? 2?r
Substitute 24? for C.
Divide both sides by 2?.
r 12
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Example 2B Continued
Find the lateral area and surface area of a right
cylinder with circumference 24? cm and a height
equal to half the radius. Give your answers in
terms of ?.
Step 2 Use the radius to find the lateral area
and surface area. The height is half the radius,
or 6 cm.
L 2?rh 2?(12)(6) 144? cm2
Lateral area
S L 2?r2 144? 2?(12)2 432? in2
Surface area
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Check It Out! Example 2 Continued
Find the lateral area and surface area of a
cylinder with a base area of 49? and a height
that is 2 times the radius.
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Example 3 Finding Surface Areas of Composite
Three-Dimensional Figures
Find the surface area of the composite figure.
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Example 3 Continued
The surface area of the rectangular prism is
.
.
Two copies of the rectangular prism base are
removed. The area of the base is B 2(4) 8 cm2.
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Example 3 Continued
The surface area of the composite figure is the
sum of the areas of all surfaces on the exterior
of the figure.
S (rectangular prism surface area)
(triangular prism surface area) 2(rectangular
prism base area)
S 52 36 2(8) 72 cm2
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Check It Out! Example 3
Find the surface area of the composite figure.
Round to the nearest tenth.
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Check It Out! Example 3 Continued
Find the surface area of the composite figure.
Round to the nearest tenth.
The surface area of the rectangular prism is
S Ph 2B 26(5) 2(36) 202 cm2.
The surface area of the cylinder is
S Ph 2B 2?(2)(3) 2?(2)2 20? 62.8 cm2.
The surface area of the composite figure is the
sum of the areas of all surfaces on the exterior
of the figure.
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Check It Out! Example 3 Continued
Find the surface area of the composite figure.
Round to the nearest tenth.
S (rectangular surface area) (cylinder
surface area) 2(cylinder base area)
S 202 62.8 2(?)(22) 239.7 cm2
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Example 4 Exploring Effects of Changing
Dimensions
The edge length of the cube is tripled. Describe
the effect on the surface area.
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Example 4 Continued
original dimensions
edge length tripled
S 6l2
S 6l2
6(24)2 3456 cm2
6(8)2 384 cm2
Notice than 3456 9(384). If the length, width,
and height are tripled, the surface area is
multiplied by 32, or 9.
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Check It Out! Example 4
The height and diameter of the cylinder
are multiplied by . Describe the effect on
the surface area.
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Check It Out! Example 4 Continued
original dimensions
height and diameter halved
S 2?(112) 2?(11)(14)
S 2?(5.52) 2?(5.5)(7)
550? cm2
137.5? cm2
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Lesson Quiz Part I
Find the lateral area and the surface area of
each figure. Round to the nearest tenth, if
necessary. 1. a cube with edge length 10 cm 2.
a regular hexagonal prism with height 15 in. and
base edge length 8 in. 3. a right cylinder with
base area 144? cm2 and a height that is the
radius
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Lesson Quiz Part II
4. A cube has edge length 12 cm. If the edge
length of the cube is doubled, what happens to
the surface area? 5. Find the surface area of
the composite figure.