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.
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THREE DIMENSIONAL MEASUREMENT...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
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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
_________ 2. Lateral Face
_________ 3. Base
_________ 4. Vertex
_________ 5. Height
A. Rectangle ADGH
B. HF
C. AD
D. Point B
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