Unit 3: Surface Area, Volume and Capacity - St John Brebeuf

Apprenticeship and workplace Math 11
St.John Brebeuf
Unit 3:
Surface Area, Volume
and Capacity
Mathematics
Department
Goals:
In this chapter you will
Estimate and calculate the surface area of a 3-D object.
Estimate, measure and calculate the volume of 3-D objects.
Modify your surface area and volume measurements when their
dimensions increase or decrease.
Explore the relationship between several objects and determine
how their volumes are related.
Key Terms: You will be able to define and use the following terms:
Cone
Cylinder
Dimension
Prism
Pyramid
Sphere
Surface Area
Volume
capacity
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Apprenticeship and workplace Math 11
St.John Brebeuf
3.1 Surface Area of Prisms:
‣ Net: 2-D pattern that can be folded to form a 3-D shape .
‣ Rectangular Prism: A 3-D shape with ends that are congruent
rectangles and with sides that are parallelograms
Rectangular Prism
Net of Rectangular Prism
‣ Surface Area: The Area required to cover a 3-D shape.
The surface area is calculated by adding up all the areas of
the surfaces of the net of the shape.
‣ Prism: A 3-D shape with ends that are congruent polygons and
sides that are parallelograms.
‣ Congruent Polygons: have
The same shape and size
Sides and angles in the same position.
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‣ Base: One of the parallel faces of a prism.
‣ Lateral Face: A face that connects the bases of a prism
Lateral Face
Base
Base
Example 1:
David builds a fence around his yard. To make the gate he cuts four
pieces of 2 x 4 lumber for a frame and one piece as a diagonal brace to
keep the gate from sagging. For the fence posts he uses 2 x 2 lumber
and cuts the ends at 45 , as shown below.
2 x 2 lumber
2 x 4 frame 45 each end
Garden gate
Fence posts
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a) What is the shape of each of the pieces he uses for the frame
b) What is the shape of the brace
c) What is the shape of each of the pickets
Solution:
a) __________________________________________________
b) __________________________________________________
c) ___________________________________________________
Example 2:
Nicola ships posters in a holder the shape of an equilateral triangular
prism:
96.5cm
15.2cm
13.2cm
15.2cm
a) Draw the net of the box and label the dimensions of each side.
b) Calculate the surface area of the box.
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St.John Brebeuf
Solution:
a)
b) Area of one rectangle: Area of one triangle
A = l x w A = bh
= = x
= =
Surface Area = 3 Rectangles + 2 Triangles
=
=
=
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St.John Brebeuf
Example 3:
Katie is building a child’s wagon. The wagon’s box is to be 3 feet long
and 18 inches wide. The sides of the wagon will be 10 inches high.
a) Draw a net of the wagon.
b) Calculate the surface area of the box
in square inches.
c) If 1ft 2 = 144 in 2 calculate the surface area of the
box in square feet.
d) Can she make the box from a single sheet of 4ft x 8ft plywood
Solution:
a)
b) Surface Area = area of base + area of ends + area of sides
=
=
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c)
96 in
d) 4ft x 8ft
= 48in x 96 in 48in
Complete notebook assignment page 124 # 1-6
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Apprenticeship and workplace Math 11
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3.2: Surface Area of Pyramids, Cylinders, Spheres and Cones:
Pyramid Cylinder Sphere Cone
‣ Pyramid: A 3-D shape with a base of a polygon and lateral sides
that are all triangles
‣ Cylinder: A 3-D shape with 2 circular bases that are parallel and
congruent; the side is a rectangle that is wrapped around the
circular bases at the ends.
‣ Sphere: A 3-D shape whose surface consists of points that are
all the same distance from the centre of the shape.
‣ Cone: A 3-D shape with a circular base and a vertex opposite the
base
‣ The method for calculating the Surface Area of these objects is
the same as for prisms:
Add up all the individual surfaces that make up the object
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Example1:
Calculate the surface area of the cylinder below in square feet
Diameter = 30 in
Solution:
Length= 24 in
Diameter = 30in
Length 24 in
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Example 2:
Calculate the surface area of the sphere below, whose diameter is 1.5
metres.
1.5m
Solution:
Surface Area of a Sphere = 4 r 2
=
=
Example 3:
Calculate the surface area of the composite shape below, where the
radius of the hemisphere and cone are 9m, the slant height of the cone
is 15m and the length of the cylinder is 20m.
15m
9m
9m
20m
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Solution:
SA = Area of the hemisphere + Area of cylinder + Area of Cone
= + 2 r 2 + d + rs
Why don’t we count these 2 circles
Example 4:
Use r to represent the radius of the small sphere, compare the surface
areas of the 2 spheres below:
r
3r
Sphere A
Sphere B
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Solution:
S.A. = rs
Example 4:
Maureen is estimating the cost to re-shingle her roof.
To determine the number of shingles, she must calculate the surface
area of the roof. The roof is shaped like a pyramid with each base side
having a base length of 25 ft and a slant height of 15 ft 3 in.
a) What is the total surface area she needs to cover
b) Each shingle has dimensions long by wide.
Maureen calculates the area of 1 shingle and divides the
area of the roof by the area of the shingle to determine the
number of shingles she will need.
Will she buy the right quantity of shingles using this
method
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c) Shingles are sold in bundles of (15-20) which cover 30 ft 2 .
How many bundles will she need
d) Each bundle costs $31.50. What is the total cost of the shingles
Solution: SA of a pyramid = b 2 + 4 x b x h
Why do we not count the base 2 in this example
a)
b)
c)
d)
Complete notebook assignment page 134 # 1-4
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3.3 Volume and Capacity of Prisms and Cylinders.
‣ Volume: The measure of the space a 3-D
object occupies, measured in units such as cm 3 , m 3 .
‣ Capacity: The amount a 3-D object can hold,
measured in units such as litres or gallons.
‣ Capacity and Volume are closely related. Capacity is the amount
of material that can be contained in a hollow volume.
Hollow objects have volume and capacity.
Solid objects have only volume.
Example 1
Louis sells different sizes of fish tanks in his pet store.
20cm
60cm
20cm
40cm
20cm
20cm
20cm
20cm
20cm
Tank 1
Tank 2
Tank 3
a) How much water will be needed to fill each tank
b) Look at tanks 2 & 3, how many dimensions have changes By how
much How does the volume of tank 2 compare to that of tank 3
c) Look at tanks 1 & 3, how many dimensions have changes By how
much How does the volume of tank 1 compare to that of tank 3
d) 1000cm 3 = 1 litre. Convert the 3 volumes into capacity in litres.
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Solution:
a) V 1 = l x w x h V 2 = l x w x h V 3 = l x w x h
b)
_____________________________________________________
_____________________________________________________
_____________________________________________________
c)
____________________________________________________
____________________________________________________
____________________________________________________
d) Tank 1 Tank 2 Tank 3
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Example 2.
Shapes called climbing grips are inserted into climbing walls for
climbers to grasp. Some climbing grips are made of polyester resin
that is poured into moulds of different shapes.
A mould for a cylindrical grip has a radius of 1.5 in and a height of 3 in.
The machine that pours the moulds must fill each mould.
What is the volume of the mould
Solution: Volume of a cylinder = Area of the base x height
V
= r 2 x h
Example 3.
1
2 4
1 2 4
1 2 4
a) Determine the volume of each box.
b) As the lengths are doubled, what happens to the volume Explain
your answer.
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Solution:
a) V 1 = V 2 V 3
b)
Example 4.
Engine displacement is the volume swept by the pistons in an engine’s
cylinders.
The following formula can be used to calculate engine displacement.
Engine displacement = [
] 2 (stroke)(# cylinders)
The bore is the diameter of the engine’s cylinder. The stroke is the
distance that the piston moves in the cylinder
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A 4-cylinder engine has a stroke of 82mm and a bore of 75.5mm
a) What is the engine displacement in cm 3
b) Calculate the engine displacement in litres.
c) [1 inch = 2.54cm.] What is the displacement of the engine in part a
in cubic inches
d) A car with a larger engine displacement generates more power than
one with a smaller engine displacement. How would this displacement
change if the stroke were doubled
e) How would the displacement change if the bore were doubled
Solution:
a) Engine displacement = [ ] 2 (stroke)(# cylinders)
b)
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c)
_____________________________________________________
_____________________________________________________
_____________________________________________________
d)
____________________________________________________
____________________________________________________
____________________________________________________
e)
____________________________________________________
____________________________________________________
____________________________________________________
____________________________________________________
Complete notebook assignment page 144 # 1-6
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Apprenticeship and workplace Math 11
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3.4 Volume and Capacity of Spheres, Cones and Pyramids:
Volume cone
= x area of base x height
Volume pyramid = x area of base x height
Volume sphere = x r 3
Example 1:
A tennis ball has a diameter of 6.7 cm.
a) What is the volume of the tennis ball
b) Tennis balls are commonly sold in cylindrical tubes in packs of three
per tube. What is the volume of the container
c) What is the ratio of the volume of the three balls to the volume of
the container
d) Using r to represent the radius, show that the volume of a sphere is
the volume of a cylinder that completely encloses it.
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Solution:
a) V ball = x r 3 d = 6.7cm r = 6.7/2
=
=
b) V container = r 2 h
=
c) =
d)
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Example 2:
When bulk materials such as sand, gravel or grain are poured onto a
flat surface the shape of the pile will be a cone. The angle between
the horizontal surface (the ground) and the slope of the pile is called
the angle of repose. The angle of repose of the pile remains the same
regardless of the height of the pile.
A gravel pile has a diameter of 3.5 m and a height of 1.2m
a) What is the volume of gravel in the pile
b) What is the volume of a pile with double the dimensions
( Use proportional reasoning.)
Solution:
a) Volume cone = x area of base x height d = 3.5m r = 3.5/2
b) Dimensions are doubled = x2
New Volume = 2 3 x Old Volume
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‣ Note:
If dimensions are increased by a scale factor S
New Area
= S 2 x Old Area
New Volume = S 3 x Old Volume
‣ Now do activity 3.6 on page 152 of textbook
Example 3.
A spherical tank containing heating oil has a diameter of 1.5m
a) What is the volume in m 3
b) How many litres are in the tank
c) [1 US gallon = 3.785 litres] How many US gallons does the tank
hold
Solution:
a) Volume sphere = x r 3
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b)
c)
Example 4.
A garden gnome is to be sculpted out of concrete made up of 3
irregular shapes, a cone for the hat, a hemisphere for the head and a
cylinder for the body.
a) What volume of concrete in ft 3 will be needed to sculpt 20 gnomes
[1728in 3 = 1 ft 3 ]
b) Once a gnome has been sculpted what method could be used to
verify the exact volume of concrete used
Solution:
a) V cylinder = r 2 h
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V hemisphere = x r 3
V cone =
r 2 h
Now add the volumes together and multiply by 20
Now change to cubic feet: 1728in 3 = 1 ft 3
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b)
____________________________________________________
____________________________________________________
____________________________________________________
____________________________________________________
____________________________________________________
Complete notebook assignment page 156 # 1-5
Complete Unit 2 Review page 160 # 1-10 odd
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Reflect on your learning
Now check the box that applies to you :
RED AMBER GREEN
I understand all the key terms.
I can explain the difference
between Volume and Surface Area.
I can explain the relationship
between area and surface area.
I can calculate the surface area
and volume of 3-D objects.
I can explain the difference
between volume and capacity.
I can convert between units of
volume.
I can calculate volume using tools
such as rulers, tape measures,
micrometers and calipers.
I can calculate capacity using
methods such as graduated
cylinders, cups, spoons and displacement.
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Apprenticeship and workplace Math 11
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I can describe the relationship
between volumes of cones and
cylinders with the same base and
height.
I can describe the relationship
between volumes of pyramids and
prisms with the same base and
height.
I can explain the effect a change
in dimensions has on the surface
area and volume.
I have completed all
homework assignments.
I have attended tutorials
for extra help.
I am ready to sit my
unit 3 test.
Target:
In my Unit Test I hope to achieve
%
Student’s Signature ____________________
Date__________
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