Problem
    6.1
    Problem
    6.2
    Topic 6: Investigating Volume
    for use before
    Looking for Pythagoras
    ( (
    Investigation 3)
    Volume is the amount of space enclosed in a solid. It is expressed in cubic
    units. The volume of the Empire State Building in New York City is about
    37 million cubic feet. The volume of a raisin is about cubic inch.
    a. The volume of a prism or a pyramid with a square base is determined
    by the height of the solid and the area of the base.
    A. What is the volume of a rectangular prism with a length of 8 cm, a
    width of 4 cm, and a height of 10 cm?
    B. What is the volume of a square pyramid with a height of 3 feet and
    a base with sides of 1 foot?
    C. Give two sets of possible dimensions for a rectangular prism with a
    volume of 100 in
    3
    .
    D. How does the volume of a rectangular prism change if you lay it on
    its side before taking measurements? Explain.
    The volume of a cone or a cylinder is determined by the height and
    the area of the base. The exact volume of a cone or cylinder includes the
    value .
    A. 1. What is the exact volume of a cylinder with radius 5 m and height 3 m?
    2. What is the exact volume of a cone with radius 7 in. and height 12 in.?
    Cone
    Volume ? ? Base ? height
    Cylinder
    Volume ? Base ? height
    ? p
    r
    2
    h
    h
    r
    1
    3
    ? p
    r
    2
    1
    h
    3
    p
    Volume ? Base ? height
    Volume ?
    1
    ? Base ? height
    3
    ?
    s
    2
    1
    h
    3
    Rectancle
    prism
    Square
    Pyramid
    Volume ? length ? width ? height
    ?
    lwh
    1
    8
    B. If a cone and a cylinder have the same radius and the same height, how
    many times greater is the volume of the cylinder than the cone?
    C. If the volume and height of a cone and a cylinder are the same, which
    one has the larger base?
    D. If the radius of a cone doubles and the height remains the same, what
    happens to the volume?
    Exercises
    Find the exact volume of each solid.
    1.
    2.
    3.
    4.
    5.
    How are the formulas for the volume of prisms and cylinders the
    same?
    6. How are the formulas for the volume of cones and pyramids the same?
    7. A watering can does not fit under a faucet, so Trang is using a paper
    cone to fill it with water. The cone has a radius of 1 inch and a height of
    4 inches. The can has a radius of 2 inches and a height of 8 inches. How
    many conefuls of water will it take to fill the can?
    8. The Great Pyramid of Khufu in Egypt has a square base with sides of
    about 230 meters and a height of about 146 meters. What is the
    approximate volume of the pyramid?
    9. The formula for the volume of a sphere is
    V
    ?
    r
    3
    .
    a. What is the volume of a baseball with a diameter of 2.8 inches?
    b. About many times more volume does a basketball with a radius of
    4.78 inches have than a baseball?
    10. What is the exact volume of the largest cone that can fit into a cube
    with sides of 10 inches?
    r
    4
    3
    p
    9 in.
    6 in.
    6 in.
    12 cm
    3 cm
    12 cm
    8 cm
    3 ft
    7 ft
    4 ft
    Topic 6: Investigating Volume
    Guided Instruction
    Mathematical Goals
    • Determine the volume of three-dimensional shapes.
    • Investigate the relationship of the volume of three-dimensional figures with
    the same base area and height.
    Vocabulary
    •
    volume
    Materials
    •
    Labsheet 6ACE
    Exercises 1–4
    At a Glance
    Use unit cubes in an exercise to give the students a hands-on experience
    with cubic measure. Show that the cubes can be rearranged into numerous
    configurations without the volume changing.
    The solid shapes used in this lesson are all right shapes: the bases of the
    rectangular prisms and cylinders are aligned vertically; the apexes of the
    pyramids and cones are directly above the center of the base. Even if these
    shapes leaned to one side, the same formulas for volume would apply.
    After Problem 6.1
    •
    Why doesn’t it matter which edges you call length, width, and height
    when you find the volume of a rectangular prism?
    (According to the
    Commutative Property of Multiplication it doesn’t matter in what
    order you do the multiplication.)
    •
    How do you define a rectangular prism?
    (a solid figure with six faces
    that are rectangles)
    •
    What do you call a rectangular prism whose sides are all squares?
    (cube)
    •
    What is the definition of a cubic inch?
    (The volume of a cube with sides
    of 1 inch ? 1 inch.)
    You will find additional work on volume in the grade 7 unit
    Filling and
    Wrapping.
    PACING
    1 day
    Assignment Guide
    for Topic 6
    Core
    1–9
    Advanced
    10
    Answers to Topic 6
    Problem 6.1
    A. 1.
    320 cm
    3
    B.
    1 ft
    3
    C.
    Answers may vary. Sample: 5 in. ? 5 in. ? 4
    in.; 10 in. ? 5 in. ? 2 in.
    D.
    The volume does not change if you place a
    rectangular prism on its side before
    measuring.
    Problem 6.2
    A. 1.
    75
    m
    3
    2.
    196
    in.
    3
    B.
    3 times
    C.
    the cone
    D.
    Volume increases by a factor of 4.
    Exercises
    1.
    84
    ft
    3
    2.
    768
    cm
    3
    3.
    36
    cm
    3
    4.
    108
    in.
    3
    5.
    They both multiply the area of a base times
    the height.
    6.
    They both multiply the area of a base times
    the height.
    7.
    24 conefuls
    8.
    2,574,467 m
    3
    9. a.
    11.5 in.
    3
    b.
    40 times more volume
    10.
    250p
    3
    in.
    3
    1
    3
    p
    p
    p
    p
    Name
    Date
    Class
    Labsheet 6ACE Exercises 1–4
    Topic 6
    1.
    2.
    3.
    4.
    9 in.
    6 in.
    6 in.
    12 cm
    3 cm
    12 cm
    8 cm
    3 ft
    7 ft
    4 ft
    © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.

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