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G.GMD.1 STUDENT NOTES WS #7
1
THREE DIMENSIONAL MEASUREMENT
The third dimension has a new vocabulary that we need to familiarize ourselves with before we can begin
looking at formulas or calculations.
A Solid – A three dimensional closed spatial figure.
A Polyhedron – a geometric solid with polygons as faces.
A Face of a Polyhedron – One of the polygons that form the
polyhedron. Sometimes these get called sides but the better
term is face.
An Edge – The intersection of two faces of a polyhedron.
A Vertex – The intersection of two or more edges.
THE PRISM
A prism is a polyhedron that consists of a polygonal region and its translated image in a parallel plane, with
quadrilateral faces connecting the corresponding edges. WOW…. That sound pretty technical… and I guess it
is. Let’s try to clarify that definition with a few examples.
The two congruent faces that have been translated into parallel planes are called the bases of the prism. The
faces that are not based are called the lateral faces. All of these examples are right prisms which mean the
base and lateral edges are perpendicular to each other. When working with right prisms the height of the
prism is also a lateral edge.
A common misconception is that whatever face the prism is ‘sitting’ on is the base – that IS NOT HOW THE
BASE IS DETERMINED!! The height of the prism is the perpendicular distance between the two congruent
bases. That brings up another item is about naming – we name a prism by its base shape We in these
examples we have a triangular prism, a rectangular prism and a hexagonal prism.
If the base and the lateral edges are NOT PERPENDICULAR then the prism is called an OBLIQUE PRISM. When
the prism is oblique the lateral faces are not rectangles, they are parallelograms as shown.
G.GMD.1 STUDENT NOTES WS #7
2
PRISM VOLUME – THE STACKING PRINCIPLE
A powerful technique to use when calculating the volume of a prism is stacking. For example, the stack of
CD’s is a solid made up of many square cases upon each other or the volume of paper made up of stacking
many rectangular sheets (8 ½ by 11) upon each other or money or coasters. All of these cross sections have a
height dimension but if we make that height infinitesimal small we begin again to approximate the
relationship more accurately.
A Stack of CD Cases
Cross Section: Square
A Stack of Paper
Cross Section: Rectangle
A Stack of Money
Cross Section: Rectangle
A Stack of Coasters
Cross Section: Squares
Because in a prism we have two translated congruent bases in parallel planes (the bases), all of the cross
sections are also identical to the bases. So to calculate the volume of a prism we calculate the area of the
base and then multiply it by the height of the prism – thus stacking that area on top of itself to fill in the
volume of the shape.
The stacking of congruent parallel cross sections allows us to create a formula for the volume of prism.
VolumePRISM = Bh, where B is the area of the base and h is the height of the prism.
G.GMD.1 STUDENT NOTES WS #7
3
Bonaventura Fransesco Cavalieri (1598-1647) was a disciple of Galileo and he investigated this stacking
principle. He came to define what is now known as the Cavalieri Principle:
If the areas of the cross sections of two solids by any plane parallel to a given
plane are invariably equal, then the two solids have the same volume.
In other words, if two prisms have the same height and the same base then oblique and right prisms will have
the same volumes. This is a little like the shearing technique but in the third dimension.
Volumes are equal.
Volumes are equal.
This will be helpful with volume of future shapes where we can use a simple cross section of equal area to
determine the volume of more complex solids.
THE CYLINDER
A cylinder has a lot of the same characteristics as a prism but is not a prism because of its circular base. The
cylinder and the prism both have two identical parallel bases and interesting enough they both have
rectangular lateral faces. The last point might be hard to visualize if
you think of unwrapping a soup can label or pulling the label from a
water bottle - the two dimensional form is a rectangle.
CYLINDER VOLUME – THE STACKING PRINCIPLE
The same stacking technique works great for cylinders as
well. All cross section parallel to the base are all congruent
circles and so using the same technique we are able to
determine the formula to be:
AREACYLINDER = Bh = πr2h
Cavalieri’s principle works with cylinders as well. The cross
section will be congruent circles and so the oblique cylinder
will have the same volume as the right cylinder.
G.GMD.1 WORKSHEET #7
NAME: ____________________________ Period _______
1
1. Match the following terms to the diagram.
Given the rectangular prism with face BCFE as one of its bases. Use
each value ONLY ONCE.
_________ 1. Edge
A. Rectangle ADGH
_________ 2. Lateral Face
B. HF
_________ 3. Base
C. AD
_________ 4. Vertex
D. Point B
_________ 5. Height
E. Rectangle HDCF
2. After looking at the rectangular prism to the right, a young lady in the class
raises her hand and says, “Could I use rectangle ADCB as my base instead of
rectangle BHGC?” How should the teacher respond?
3. Properly name the following prisms.
a)
b)
c)
d)
Name:
____________________
Name:
____________________
Name:
____________________
Name:
____________________
e)
f)
g)
h)
Name:
____________________
Name:
____________________
Name:
_____________________
Name:
_____________________
4. Mike doesn’t understand how volume works for a prism and Henry is trying to explain it to him. “It’s
what is inside the shape… for example, if you calculated the area of one piece of paper and then stacked
100 pieces of paper on top of each other it would create a prism and the volume would be the area of the
one piece of paper multiplied by the height of the stack.”. Mike is still confused, can you give another
example to explain this concept.
G.GMD.1 WORKSHEET #7
2
5. Cavalieri’s principle says that these two prisms have equal
volume. Explain why that is true?
6. If the Volume of the cube is (4)(4)(4) =
64 cm3, what is the volume of the oblique
prism if it has been tilted at 60°°?
7. Jenny says that the two prisms DO NOT have the same volume because the cross sections are not the
same. Renee disagrees; she says that it isn’t the shape that has to be the same it is the area. Renee thinks
they have the same volume. Who is right and why?
8. An enclosed glass box contains 1620 in3 of water. When the glass box is tilted on its side the water shifts
places. What is the relationship of the water before and after the tilting? What is the height of the water
when the box is tiled upright?
G.GMD.1 WORKSHEET #7
3
9. Randy says “Cylinder volume is easy – it is done the same way as a prism except its base is a circle.” What
does Randy mean by this?
10. Ryllie looks at the two cylinders below and says that the oblique cylinder has less volume because it is at
an angle. The volume for the right cylinder is V = π(3)2(h) and the volume for the oblique cylinder is V =
π(3)2(sin 73°°)(h) because it is at an angle. Is she correct?
11. Jared wants to test out a new theory….. Instead of having the
cross area sections the same as Cavalieri suggested he wants to half
the radius of one cross section and then double the height to make
up for it. He believes because he divided the radius by 2 but
doubled the height that the volumes should be equal. Is he
correct? Explain.
12. A rectangular prism and a cylinder have the same height and the same volume. What is the length of
the side of the prism’s square base?
*o*r,
G.GMD.7 WORKSHEET #7
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1. Match the following terms to the diagram.
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2. After looking at the rectangular prism to the right, a young lady in the class
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rectangle BHGC?" How should the teacher respond?
3. Properly name the following prisms.
a)
d)
b)
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4. Mike doesn't understand how volume works for a prism and
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what is inside the shape... for example, if you calculated the area of one piece of paper and then stacked
100 pieces of paper on top of each other it would create a prism and the volume would be the area of the
one piece of paper multiplied by the height of the stack.". Mike is still confused, can you give another
example to explain this concept.
G.GMD.7 WORKSHEET #7
5. Cavalieri's principle says that these two prisms have equal
volume. Explain why that is true?
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5. lf the Volume of the cube is (4X4X4) =
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prism if it has been tilted at 60"?
two prisms DO NOT have the same volume because the cross sections are not the
same. Renee disagrees; she says that it isn't the shape that has to be the same it is the area. Renee thinks
they have the same volume. Who is right and why?
l|"i7. Jenny says that the
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8. An enclosed glass box contains 1520 in3 of water. When the glass box is tilted on its side the water shifts
places. What is the relationship of the water before and after the tilting? What is the height of the water
when the box is tiled upright?
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G.GMD.I
#7
WORKSHEET
3
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11. Jared wants to test out a new theory..... lnstead of having the
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the radius of one cross section and then double the height to make
up for it. He believes because he divided the radius by 2 but
doubled the height that the volumes should be equal. ls he
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