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Math Help - Calculation of the area of a circle segment in cubic inches

  1. #1
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    Calculation of the area of a circle segment in cubic inches

    Calculation of the area of a circle segment in cubic inches

    I am trying to calculate the amount of displacement a barrel partially submerged has in cubic inches.

    barrel is 20x30.

    I am able to calculate the chord length and height at a given circle center angle but when attempting to convert to cubic inches, the values are too large.example: using barrel dimensions above and a radius of 10 and a center angle of 135, i get a chord length of 18.48, height of 6.17 and area of 82.45.

    When i attempt to get the volume of the area by 82.45 x 30 = 2473.03 cubic inches.

    2473/231 (cubic inches per gallon of water) = 10.71 gallons.
    10.71 x 8.3 (#'s/gallon water) = 89.31 displacement pounds.

    I believe the area of the segment doesn't represent square inches but I don't know why or what it represents instead. A full 30 gallon barrel submerged should only displace about 250 pounds. My calculations show a full barrel displacing about 340 pounds so an assumption I have made about the area calculation/cubic volume result is wrong and I don't know why.

    The volume calculation takes the area of the segment here:
    82.45 =((10^2)/2)*(2.36-SIN(2.36)) (2.36 is radians = 135 degrees)

    The volume calculation takes the area, thought to be square inches, multiplied by the length of the barrel (30).
    82.45 * 30 = 2573.63 cubic inches

    Gallon calcullation
    2573.63 / 231 = 10.71 (231 is # cubic inches per gallon water)

    Pounds Calculator (# gallons x wt per gallon)
    10.71 * 8.3 = 89.31 pounds displacent. (8.3 pounds per gallon of water.)
    Last edited by daverimmer; June 12th 2011 at 11:57 AM. Reason: clarification and formatting
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  2. #2
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    You seem to be mixing dimensions: the area of a circle segment would not be measured in cubic inches.
    What are the dimensions of the barrel? Is 20 the height or the radius of the barrel?
    Is the barrel laying horizontal in the water, or vertical? How deep in the water?
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  3. #3
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    Hi daverimmer,
    Post all givens and clearly state the questions




    bjh
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  4. #4
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    Givens

    Barrel

    20 inches wide
    30 inches tall
    Radius 10 inches

    Barrel lies horizontal in the water

    The portion in the water is 20 in - 6.17 in ( chord height )

    Barrel lies between two joists

    The portion above water, settled between joists, is the volume I wish to calculate, as, being above water, does not provide displacement.

    I assumed that the area of the segment is square inches and multiplying this area by the length would give a cubic volume value.

    So, here is my question: in the area equation written in my first post, what are the units expressed by the 82.45 result? If not square inches then what?

    I need to know how to calculate the volume of the portion of the barrel which is above water if the chord height 6.17, chord length is 18.48, radius 10, center angle of 135 degrees, length 30 inches and area of 82.45.

    Barrel contains total of 55 gallons
    Last edited by daverimmer; June 12th 2011 at 08:53 PM. Reason: Clarification
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  5. #5
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    I never said the area of a circle was expressed in cubic inches. I said I assumed the area was expressed as square inches.
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  6. #6
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    Never mind, figured it out, h is 20 not 30. Thanks for the responses.

    Dave
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  7. #7
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    Re: Calculation of the area of a circle segment in cubic inches

    Hi daverimmer,
    Volume of drum above water is the volume of cylinder less volume of segmental shape metal thickness neglected
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