Transcript Document

Lesson 59
Surface area and Volumes of Prisms
Warm Up
Find the perimeter and area of
each polygon.
1. a rectangle with base 14 cm and height
9 cm P = 46 cm; A = 126 cm2
2. a right triangle with 9 cm and 12 cm
legsP = 36 cm; A = 54 cm2
3. an equilateral triangle with side length
6 cm
Objectives
Learn and apply the formula for the
surface area of a prism.
Vocabulary
lateral face
lateral edge
right prism
oblique prism
altitude
surface area
lateral surface
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.
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.
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.
The surface area of a right rectangular prism with
length ℓ, width w, and height h can be written as
S = 2ℓw + 2wh + 2ℓh.
Caution!
The surface area formula is only true for right
prisms. To find the surface area of an oblique
prism, add the areas of the faces.
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.
L = Ph
P = 2(9) + 2(7) = 32 ft
= 32(14) = 448 ft2
S = Ph + 2B
= 448 + 2(7)(9) = 574 ft2
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.
L = Ph
= 30(20) = 600 cm2 P = 3(10) = 30 cm
S = Ph + 2B
The base area is
Check It Out! Example 1
Find the lateral area and surface area of a cube
with edge length 8 cm.
L = Ph
= 32(8) = 256 cm2
P = 4(8) = 32 cm
S = Ph + 2B
= 256 + 2(8)(8) = 384 cm2
Example 3: Finding Surface Areas of Composite
Three-Dimensional Figures
Find the surface area of the composite figure.
Example 3 Continued
The surface area of the rectangular prism is
.
A right triangular prism is added to the
rectangular prism. The surface area of the
triangular prism is
.
Two copies of the rectangular prism base are
removed. The area of the base is B = 2(4) = 8 cm2.
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
Example 4: Exploring Effects of Changing Dimensions
The edge length of the cube is tripled. Describe
the effect on the surface area.
Example 4 Continued
24 cm
original dimensions:
S = 6ℓ2
= 6(8)2 = 384 cm2
edge length tripled:
S = 6ℓ2
= 6(24)2 = 3456 cm2
Notice than 3456 = 9(384). If the length, width, and
height are tripled, the surface area is multiplied by 32,
or 9.
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?
The surface area is multiplied by 4.
5. Find the surface area of the composite figure.
S = 3752 m2
Objectives
Learn and apply the formula for the
volume of a prism.
Vocabulary
volume
The volume of a three-dimensional figure is the
number of nonoverlapping unit cubes of a given size
that will exactly fill the interior.
Cavalieri’s principle says that if two threedimensional figures have the same height and have
the same cross-sectional area at every level, they
have the same volume.
A right prism and
an oblique prism
with the same base
and height have the
same volume.
Example 1A: Finding Volumes of Prisms
Find the volume of
the prism. Round to
the nearest tenth, if
necessary.
Volume of a right rectangular prism
V = ℓwh
= (13)(3)(5) Substitute 13 for ℓ, 3 for w, and 5 for h.
= 195 cm3
Example 1B: Finding Volumes of Prisms
Find the volume of a cube with edge length 15
in. Round to the nearest tenth, if necessary.
V = s3
= (15)3
= 3375 in3
Volume of a cube
Substitute 15 for s.
Example 1C: Finding Volumes of Prisms
Find the volume of the right
regular hexagonal prism. Round
to the nearest tenth, if
necessary.
Step 1 Find the apothem a of the base. First draw a
right triangle on one base. The measure of the angle
with its vertex at the center is
.
Example 1C Continued
Find the volume of the right
regular hexagonal prism. Round
to the nearest tenth, if
necessary.
So the sides are in ratio
.
The leg of the triangle is half the
side length, or 4.5 ft.
Solve for a.
Step 2 Use the value of a to find the base area.
P = 6(9) = 54 ft
Example 1C Continued
Find the volume of the right
regular hexagonal prism. Round
to the nearest tenth, if
necessary.
Step 3 Use the base area to find the volume.
Check It Out! Example 1
Find the volume of a triangular prism with a
height of 9 yd whose base is a right triangle with
legs 7 yd and 5 yd long.
Volume of a triangular prism
Example 2: Recreation Application
A swimming pool is a rectangular prism.
Estimate the volume of water in the pool in
gallons when it is completely full (Hint: 1 gallon
≈ 0.134 ft3). The density of water is about 8.33
pounds per gallon. Estimate the weight of the
water in pounds.
Example 2 Continued
Step 1 Find the volume of the swimming pool in
cubic feet.
V = ℓwh = (25)(15)(19) = 3375 ft3
Step 2 Use the conversion factor
the volume in gallons.
to estimate
Example 2 Continued
Step 3 Use the conversion factor
estimate the weight of the water.
to
 209,804 pounds
The swimming pool holds about 25,187 gallons. The
water in the swimming pool weighs about 209,804
pounds.
Check It Out! Example 2
What if…? Estimate the volume in gallons and
the weight of the water in the aquarium if the
height were doubled.
Step 1 Find the volume of
the aquarium in cubic feet.
V = ℓwh = (120)(60)(16) = 115,200 ft3
Check It Out! Example 2 Continued
What if…? Estimate the volume in gallons and
the weight of the water in the aquarium if the
height were doubled.
Step 2 Use the conversion
factor
to estimate the
volume in gallons.
Check It Out! Example 2 Continued
What if…? Estimate the volume in gallons and
the weight of the water in the aquarium if the
height were doubled.
Step 3 Use the conversion
factor
to estimate
the weight of the water.
Check It Out! Example 2 Continued
What if…? Estimate the volume in gallons and
the weight of the water in the aquarium if the
height were doubled.
The swimming pool holds
about 859,701 gallons. The
water in the swimming pool
weighs about 7,161,313
pounds.
Check It Out! Example 4
The length, width,
and height of the
prism are doubled.
Describe the effect
on the volume.
original dimensions:
dimensions multiplied by 2:
V = ℓwh
= (1.5)(4)(3)
V = ℓwh
= (3)(8)(6)
= 18
= 144
Doubling the dimensions increases the volume by
8 times.
Example 5: Finding Volumes of Composite ThreeDimensional Figures
Find the volume of the composite
figure. Round to the nearest tenth.
The volume of the rectangular
prism is:
V = ℓwh = (8)(4)(5) = 160 cm3
The base area of the
The volume of the regular
regular triangular prism is: triangular prism is:
The total volume of the figure is the sum of the volumes.