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Math Help - Volume of Pyramids and Cones

  1. #1
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    Volume of Pyramids and Cones

    B = Area of the Base
    h = Height of the figure

    Volume Formula:

    Pyramid: 1/3Bh
    Cones: 1/3Bh


    So what this is telling me
    :
    A pyramid fits perfectly inside a prism and takes up 1/3 of the volume.

    A cone fits perfectly inside a cylinder, and takes up 1/3 of the volume.

    My Question:
    How do I know this? How was this figured out?

    I am trying to find a way to explain this to my 7th grade students without forcing them to take my word that this is true. I want them to KNOW this is true.

    My first instinct was to create a model to represent it but that is easier said than done.
    The only thing I have been able to tell them so far is that it is just like area of a triangle, but with 3d shapes. However, unlike triangles, I have been unable to show them.
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  2. #2
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    Geometrically, you can fit the pyramids into a prism.
    I can show you how to do this later if you wish.

    If you have a cone and pyramid with equal base areas and equal heights,
    their volumes are equal.
    This is because the cross-sectional areas of both at any intermediate height are equal.
    (Imagine they were both empty containers and fill up with rain to the exact same height)

    Calculus (calculation of volume) can also give a proof.
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  3. #3
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    Something fun to do is to get some prisms and their corresponding pyramids. Fill the pyramid with water and pour the water into the prism. Show them that it takes pouring the water from the pyramid into the prism three times to fill the prism.
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    "Geometrically, you can fit the pyramids into a prism.
    I can show you how to do this later if you wish."

    I would love that. My primary purpose is to show my students that the formulas have meaning and aren't made up.
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    Quote Originally Posted by Prove It View Post
    Something fun to do is to get some prisms and their corresponding pyramids. Fill the pyramid with water and pour the water into the prism. Show them that it takes pouring the water from the pyramid into the prism three times to fill the prism.
    Any idea where I can purchase some hollowed prisms and pyramids?
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    Quote Originally Posted by Jman115 View Post
    Any idea where I can purchase some hollowed prisms and pyramids?
    I expect your school would have a set somewhere - speak to your head of maths. Otherwise, look through the educational supply catalogues that will come to your school and urge your head of maths to spend some money on them.

    Another alternative is to have students build their own prisms and pyramids from cardboard (you'll need to show them what the net looks like though, or better yet, photocopy the nets onto the cardboard so they are ready-to-cut) and then use rice instead of water...
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    None of the catalogues I have received have those kind of manipulatives. The school itself is dirt poor, so we have next to no manipulatives or money.

    But the cardboard idea may work. Thanks!
    Last edited by Jman115; February 7th 2011 at 11:33 AM.
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    Quote Originally Posted by Jman115 View Post
    None of the catalogues I have received have those kind of manipulatives. The school itself is dirt poor, so we have next to no manipulatives or money.

    But the cardboard idea may work. Thanks!
    Just remember that if you have the students make their own prisms and pyramids, the activity will take a LONG time. If you are short on time, make them yourself and just give a demonstration to the class.
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  9. #9
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    On top of it!
    Already made my own.
    I only have 43 minute periods and my students would take forever to make their own. That would be a nightmare!
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    Quote Originally Posted by Jman115 View Post
    On top of it!
    Already made my own.
    I only have 43 minute periods and my students would take forever to make their own. That would be a nightmare!
    Good to hear - just make sure that you've made them strong enough so that they don't leak or collapse when you're trying to pour the rice, that'd be a nightmare (and you'll end up facing the wrath of the school janitor HAHAHA)
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  11. #11
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    Each of the green pyramids in the attachment have a volume of a third of (base area)(perpendicular height)

    V=\frac{1}{3}{\pi}r^2h

    In fig.B, base area is (0.5){\pi}r(2r), height is h

    In fig.C, base area is (0.5){\pi}r(2r), height is h

    In fig.D, base area is (0.5){\pi}rh, height is 2r

    Hence the green pyramids have equal volume
    and are therefore each one third of the volume of the red prism, whose volume is {\pi}r^2h

    Each pyramid volume is \displaystyle\frac{1}{3}{\pi}r^2h


    Comparing to cones and cylinders...

    We can say that if cross-sectional areas are equal at any height, then the volumes are equal.

    Then we can compare a cone with a pyramid, both of height h
    and base area {\pi}r^2
    Attached Thumbnails Attached Thumbnails Volume of Pyramids and Cones-prisms.jpg  
    Last edited by Archie Meade; February 11th 2011 at 11:29 AM.
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