12.2
Surface Area of Prisms and
Cylinders
Goal
Your Notes
p Find the surface areas of prisms and cylinders.
VOCABULARY
Prism
Lateral faces
Lateral edges
Surface area
Lateral area
Net
Right prism
Oblique prism
Cylinder
Right cylinder
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Lesson 12.2 • Geometry Notetaking Guide
329
12.2
Surface Area of Prisms and
Cylinders
Goal
Your Notes
p Find the surface areas of prisms and cylinders.
VOCABULARY
Prism A prism is a polyhedron with two congruent
faces, called bases, that lie in parallel planes.
Lateral faces The lateral faces of a prism are
parallelograms formed by connecting the
corresponding vertices of the bases.
Lateral edges The lateral edges of a prism are the
segments connecting the corresponding vertices
of the bases.
Surface area The surface area of a polyhedron is
the sum of the areas of its faces.
Lateral area The lateral area of a polyhedron is the
sum of the areas of its lateral faces.
Net A net of a polyhedron is a two-dimensional
representation of the faces of a polyhedron.
Right prism In a right prism, each lateral edge is
perpendicular to both bases.
Oblique prism An oblique prism is a prism with
lateral edges that are not perpendicular to the
bases.
Cylinder A cylinder is a solid with congruent
circular bases that lie in parallel planes.
Right cylinder In a right cylinder, the segment
joining the centers of the bases is perpendicular to
the bases.
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Lesson 12.2 • Geometry Notetaking Guide
329
Your Notes
THEOREM 12.2: SURFACE AREA OF A RIGHT PRISM
The surface area S of a right prism is the
sum of the base areas and the lateral area:
P
S 5 2B 1
h
where B is the area of a base, P is the
perimeter of a base, and h is the height.
B
Find the surface area of a right prism
Example 1
Find the surface area and lateral area
of the right prism.
6 in.
Solution
Step 1 Find the area of the triangular
base.
6 in.
8 in.
6 in.
Use the formula for the area of
an equilateral triangle.
1
}
Ï3 (s2)
Area of base B 5 }
4
}
1 }
5 } Ï 3 ( )2 5 Ï3 in2
6 in.
6 in.
4
Perimeter P 5 3( ) 5
6 in.
in.
Step 2 Use the formula for surface area for a right prism.
}
S 5 2B 1 Ph 5 2( Ï3 )
( )<
The surface area of the prism is about
inches.
The lateral area is Ph 5 (
)( ) 5
square
square inches.
Checkpoint Complete the following exercise.
1. Find the surface area and lateral area of a right
rectangular prism with height 5 feet, length 11 feet,
and width 4 feet.
330 Lesson 12.2 • Geometry Notetaking Guide
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Your Notes
THEOREM 12.2: SURFACE AREA OF A RIGHT PRISM
The surface area S of a right prism is the
sum of the base areas and the lateral area:
P
S 5 2B 1 Ph
h
where B is the area of a base, P is the
perimeter of a base, and h is the height.
B
Find the surface area of a right prism
Example 1
Find the surface area and lateral area
of the right prism.
6 in.
Solution
Step 1 Find the area of the triangular
base.
6 in.
8 in.
6 in.
Use the formula for the area of
an equilateral triangle.
1
}
Ï3 (s2)
Area of base B 5 }
4
}
1 }
5 } Ï 3 (6)2 5 9Ï3 in2
6 in.
4
6 in.
6 in.
Perimeter P 5 3(6) 5 18 in.
Step 2 Use the formula for surface area for a right prism.
}
S 5 2B 1 Ph 5 2(9 Ï3 ) 1 18(8) < 175.18
The surface area of the prism is about 175.18 square
inches.
The lateral area is Ph 5 (18)(8) 5 144 square inches.
Checkpoint Complete the following exercise.
1. Find the surface area and lateral area of a right
rectangular prism with height 5 feet, length 11 feet,
and width 4 feet.
The surface area is 238 ft2.
The lateral area is 150 ft2.
330 Lesson 12.2 • Geometry Notetaking Guide
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Your Notes
THEOREM 12.3: SURFACE AREA OF A RIGHT
CYLINDER
The surface area S of a right cylinder is
the sum of the base areas and the
lateral area:
B 5 pr 2
C 5 2pr
h
S 5 2B 1 Ch 5
,
where B is the area of a base, C is the
circumference of a base, r is the radius
of a base, and h is the height.
Example 2
r
Find the surface area of a cylinder
Find the surface area of the right cylinder.
Solution
Each base has a radius of
meters,
and the cylinder has a height of
meters.
S 5 2πr 2 1 2πrh
5
π1
5
π
4m
Surface area of a cylinder
)2 1 2π(
5 2π (
5m
)(
π
)
Substitute.
Simplify.
Add.
<
Use a calculator.
The surface area is about
square meters.
Checkpoint Complete the following exercise.
2. Find the surface area and lateral area of a right
cylinder with height 9 centimeters and radius 6
centimeters. Round your answer to two decimal
places.
Copyright © Holt McDougal. All rights reserved.
Lesson 12.2 • Geometry Notetaking Guide
331
Your Notes
THEOREM 12.3: SURFACE AREA OF A RIGHT
CYLINDER
The surface area S of a right cylinder is
the sum of the base areas and the
lateral area:
B 5 pr 2
C 5 2pr
h
S 5 2B 1 Ch 5 2πr 2 1 2πrh ,
where B is the area of a base, C is the
circumference of a base, r is the radius
of a base, and h is the height.
Example 2
r
Find the surface area of a cylinder
Find the surface area of the right cylinder.
Solution
Each base has a radius of 4 meters,
and the cylinder has a height of 5 meters.
S 5 2πr 2 1 2πrh
5m
4m
Surface area of a cylinder
5 2π ( 4 )2 1 2π( 4 )( 5 )
Substitute.
5 32 π 1 40 π
Simplify.
5 72 π
Add.
< 226.19
Use a calculator.
The surface area is about 226.19 square meters.
Checkpoint Complete the following exercise.
2. Find the surface area and lateral area of a right
cylinder with height 9 centimeters and radius 6
centimeters. Round your answer to two decimal
places.
The surface area is about 565.49 cm2.
The lateral area is 339.29 cm2.
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Lesson 12.2 • Geometry Notetaking Guide
331
Your Notes
Example 3
Find the height of a cylinder
Find the height of the right cylinder
shown, which has a surface area of
198.8 square millimeters.
h
2.8 mm
Solution
Substitute known values in the formula
for the surface area of a right cylinder
and solve for the height h.
S 5 2πr 2 1 2πrh
)2 1 2π(
5 2π(
5
2
Surface area
of a cylinder
π1
)h
Substitute.
πh Simplify.
π5
πh
Subtract
<
πh
Simplify. Use
a calculator.
<h
π
from each
side.
Divide
each side
by
π.
The height of the cylinder is about
millimeters.
Checkpoint Complete the following exercise.
3. Find the radius of a right cylinder with height
5 inches and surface area 168π square inches.
Homework
332 Lesson 12.2 • Geometry Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 3
Find the height of a cylinder
Find the height of the right cylinder
shown, which has a surface area of
198.8 square millimeters.
h
2.8 mm
Solution
Substitute known values in the formula
for the surface area of a right cylinder
and solve for the height h.
S 5 2πr 2 1 2πrh
Surface area
of a cylinder
198.8 5 2π( 2.8 )2 1 2π( 2.8 )h
Substitute.
198.8 5 15.68 π 1 5.6 πh Simplify.
198.8 2 15.68 π 5 5.6 πh
149.54 < 5.6 πh
8.5 < h
Subtract
15.68 π
from each
side.
Simplify. Use
a calculator.
Divide
each side
by 5.6 π.
The height of the cylinder is about 8.5 millimeters.
Checkpoint Complete the following exercise.
3. Find the radius of a right cylinder with height
5 inches and surface area 168π square inches.
7 in.
Homework
332 Lesson 12.2 • Geometry Notetaking Guide
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