1. No category

G.GMD.4 STUDENT NOTES & PRACTICE WS #1
1
REVOLING 2-D SHAPES TO GET VOLUME
Instead of stacking congruent cross sections to form the volume of a solid another technique is to rotate is
about an axis. Thinking about a revolving door and how it swings (rotates) about the center creating a
cylindrical shape.
Revolving Door
Cross Section of Door
Imagine as the door revolves it filling the
entire space of the cylinder
So a different way to fill space and create volume is to rotate figures (2-D shapes) about an axis and in doing so
fill space and create volume. This technique gets used quite a bit in higher mathematics to determine the
volume of a shape. In calculus, we find ourselves finding the area under the curve for all kinds of different
shapes. This is great preparation for these ideas.
Determine the 3-D solid that would be formed by rotating the cross section about line m.
1.
2.
3.
Answers:
1.
2.
3.
G.GMD.4 WORKSHEET #1
NAME: ____________________________ Period _______
1. Describe the solid that is formed by rotating each of these figures about line m and sketch it.
a)
b)
Name/Description
e)
c)
Name/Description
f)
Name/Description
d)
Name/Description
g)
Name/Description
Name/Description
h)
Name/Description
Name/Description
2. Determine the rotational cross section
a) A cylinder has a cone subtracted from its volume.
What does the cross section look like?
b) A hemisphere on a cylinder.
What does the cross section look like?
1
G.GMD.4 WORKSHEET #1
2
3. A potter creates pots and bowls using a pottery wheel. The wheel spins and the potter shapes the clay.
From these three pictures, create the rotational cross section.
a)
b)
c)
4. a) Use the rotational cross section to sketch the solid.
SKETCH OF SOLID
b) Use the rotational cross section to sketch the solid.
SKETCH OF SOLID
G.GMD.4 WORKSHEET #2
NAME: ____________________________ Period _______
1. What does the rotational cross section (a rectangle) have to do with the volume (cylinder)?
What is the height of the
cylinder?
What is the radius of the
cylinder?
_____________
_____________
Use the volume formula, V = π r 2 h to calculate the
volume.
Volume = __________________
Now let us see if there is a relationship between the cross
section and the volume.
What is the height of the rectangle? _____________
What is the width of the rectangle? _____________
What is the area of the rectangle? _____________
What is the relationship between the area of the rectangle and the volume of the cylinder?
2. What does the rotational cross section (a rectangle) have to do with the volume (cylinder)?
What is the height of the
cylinder?
What is the radius of the
cylinder?
_____________
_____________
Use the volume formula, V = π r 2 h to calculate the volume.
Volume = __________________
Now let us see if there is a relationship between the cross
section and the volume.
What is the height of the rectangle? _____________
What is the width of the rectangle? _____________
What is the area of the rectangle? _____________
What is the relationship between the area of the rectangle and the volume of the cylinder?
1
G.GMD.4 WORKSHEET #2
2
3. After doing the two previous problems a student suggests that to find the volume of a rotating rectangle
all you have to do is multiply the area of the rectangle (rh) times the circumference (2π
πr) of the circular path
that it travels. What do you think of this suggestion? Provided are the values from question #1, use the
values to prove or disprove this conjecture.
Example #1
Area of Rectangle = 12 cm2
Radius = 4 cm
Volume = 48π cm3
(Area of Rectangle)(Circumference) = Volume ??
4. Another student thinks that it is isn’t that easy because the circumference is different for each point, the
ones closer to the line will move on a small circular path (small circumference) and the ones further away
will move on a larger circular path (bigger circumference). What do you think of this suggestion?
5. They both have elements of truth… Yes you can multiply the area x the circumference… and yes the
circumference differs for each point so use let us use the ‘average’ circumference for the points…. What is
the radius of the ‘average’ circumference? And how might this ‘average’ circumference helps us calculate
the volume?
V = rh • 2π raverage
I
t
G.GMD.4 WORKSHEET #7
NAME:
1i
Period
1. Describe the solid that is formed by rotating each of these figures about line m and sketch it.
b)
a)
d)
c)
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Name/Description
Name/Description
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Name/Description
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Name/Description
he"'^!/
f)
h)
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ffi
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Name/Description
Cg
Name/Description
l,wA. ud/ h" L!
Name/Description
Name/Description
Csu*- &
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2. Determine the rotational cross section
a) A cylinder has a cone subtracted from its volume.
What does the cross section look like?
a
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b)A hemisphere on a cylinder.
What does the cross section look like?
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G,GMD.
2
WORKSHEET #1
potter shapes the clay'
pottery wheer. The wheer spins and the
a
using
bowrs
and
pots
3. A potter creates
the rotational cross section'
From these three pictures, create
to sketch the solid'
4. a) Use the rotatibnal cross section
to sketch the solid'
b) Use the rotational cross section
SKETCH OF SOLID
SKETCH OF SOLID
t/
NAME: t\U_
G.GMD.4WORK'HEET #2
1. What does the
period
rotational cross section (a rectangle) have to do with the volume (cylinder)?
What is the height of the
cylinder?
What is the radius of the
cylinder?
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Use the volume formula, V = 7tr2h
to calculate the
volume.
v=w)(trb)
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Volume =
Now let us see if there is a relationship between the cross
section and the volume.
What is the height of the rectangle?
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What is the width of the rectangle?
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What is the area of the rectangle?
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What is the relationship between the area of the rectangle
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2. What does the rotational cross section (a rectangle) have to do with the volume (cylinder)?
What is the height of the
What is the radius of the
cylinder?
cylinder?
Use the volume formula,
Volume
V = xr2h to calculate the volume.
=
Now let us see if there is a relationship between the cross
section and the volume.
What is the height of the rectangle?
What is the width of the rectangle?
What is the area of the rectangle?
What is the relationship between the area of the rectangle and the volume of the cylinder?
G.GMD.4 WORKSHEET #2
3. After doing the two previous problems a student suggests that to find the volume of a rotating rectangle
all you have to do is multiply the area of the rectangle (rh) times the circumference (2rcrl of the circular path
that it travels. What do you think of this suggestion? Provided are the values from question #1, use the
values to prove or disprove this coniecture.
Example
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Area of Rectangle = 12 cmz
(tgg_of Rectagg.!9X9tr*rf*g)
Radius = 4 cm
= Volume ??
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that it is isn't that easy because the circumference is different for each point, the
ones closer to the line will move on a small circular path (small circumference) and the ones further away
wil! move on a larger circular path (bigger circumference). What do you think of this suggestion?
4. Another student thinks
5. They both have elements of truth... Yes you can multiply the area x the circumference... and yes the
circumference differs for each point so use let us use the 'average' circumference for the points.... What is
the radius of the 'average' circumference? And how might this 'average' circumference helps us calculate
the volume?
f.
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