11'111'3: Volume

1
11.1-11.3 Volume

• Chapter 11 Volume
• Geometry CP

2
Volume

• Volume is the amount of space contained within an
  object (a solid).
• Solid three dimensional figure that has length,
  width and height.
• Polyhedron A solid formed by polygons that
  enclose a single region of space
• Recall
• Face Flat polygonal surface of polyhedron
• Edge The segment where two faces intersect
• Vertex A point of intersection of three or more
  edges

3
More Background

• Right prism A prism whose lateral faces are
  rectangles so the lateral edges are perpendicular
  to the bases
• Right cone The line segment connecting the
  vertex of a cone with the center of its base is
  perpendicular to the base
• Right cylinder The axis of the cylinder is
  perpendicular to the bases

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Volume Measurement
1
1
1
Length 1 unit
Volume 1 cubic unit

• Volume is the measure of the amount of space
  contained in a solid.
• Cubic units measure volume
• Ex cubic inches (in3)
• Ex Cubic meters (m3)
• Ex Cubic centimeters (cm3)
• Why?
• It is the number of unit cubes that can be
  arranged completely to fill the space within the
  object

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Investigation

• Review the handout
• The figures are right rectangular prisms
• Find the volume of each figure
• In other words, how many cubes measuring 1 cm on
  an edge will fit into each solid?

Can you determine the formula we will use for the
volume of any right rectangular prism?
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Conjecture

• For a right rectangular prism
• V BH
• Where
• B area of the base
• H height of the solid

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Example 1
4
5
6

• Find the volume
• V (area of base)(h)
• What is the base?
• (6)(4) 24 in2
• V (24in2)(5)
• V 120 in3

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Example 2

• Find the cubic yards of cement needed for a 60ft
  long driveway that is 8 inches thick and 20ft
  wide.

First, change in into ft
12 1
V BH where B area of base
V (20)(60)(.67)
3 ft 1 yd
V (1200sq ft)(.67)
804 cubic ft
27 ft3 1 yd3
Now we need to convert ft3 into yds3
804 ft3 ? 27 ft3 29.8yd3
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Conjecture

• Volume of a right cylinder
• V BH
• Where B area of the base
• But what is the area of the base?
• ?r2
• Therefore
• V (?r2)H

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Example 3

• Find the volume of the cylinder that has a base
  with a radius of 6 inches and a height of 7
  inches
• V BH
• V (?62)(7)
• V 36in2?(7)
• V 252?in3

11
Example 4

• A stack of 150 1-inch washers are 9in tall. If
  the inside hole of the washers has a diameter of
  3/4 inches, find the volume of metal in one
  washer.

volume of stack including inside hole
V pr2h
volume of all washers
p(.5in)29in
7.07 - 3.98
7.07in3
3.09in3
volume of inside hole (stacked)
volume of one washer
V pr2h
p(.375in)29in
3.09 ? 150
3.98in3
.02in3
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Homework

• P541-542
• 1-6, 9-11, 13, 14