Volume of a Trapezoidal Solid
Date: 11/15/2000 at 23:08:25 From: Greg Subject: Area/volume of a trapezoid I have a volume that is 75 ft long. The front of the figure is 57 ft wide, 35 ft high, while the rear is 72 ft wide, 12 ft high. I know how to figure the volume L*W*H and the area L*W, but how do I account for the slope of the ceiling and the opposite widths?
Date: 11/16/2000 at 08:56:15 From: Doctor Peterson Subject: Re: Area/volume of a trapezoid Hi, Greg. It sounds like your shape can be thought of as a 57x35 foot rectangle at the front, joined by planes to a 72x12 foot rectangle at the back, 75 feet away. Here is a formula for the volume of that shape, which I will draw as a frustum-like figure with rectangular top and bottom rather than front and back (since that is the form in which I have most often dealt with it): a2 +---------------+ b2/ / \--------- +---------------+ \ | / | \ | /.................|.... + | h / | / | / | / | / | /b1 ------ / | / / |/ +-----------------------+ a1 V = [a1*b1 + a2*b2 + (a1*b2 + a2*b1)/2] * h/3 Here the bottom is a1 x b1, and the top is a2 x b2, with the a's parallel and the b's parallel (this is important). An alternative version of the formula, using the average length and width, is: V = [a1*b1 + a2*b2 + 4((a1+a2)/2 * (b1+b2)/2)] * h/6 \___/ \___/ \___________________/ area of area of area of bottom top middle rectangle rectangle rectangle The "middle" rectangle has sides that are the average of the sides of the top and bottom rectangles: a2 +---------------+ b2/ / \--------- +---------------+ + | / | / \ | /.................|./.. + | h / |/ / | +-------------------+ / | / (a1+a2)/2 | /b1 ------ / | / / |/ +-----------------------+ a1 In your case, a1 = 57 b1 = 35 a2 = 72 b2 = 12 h = 75 - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 08/27/2001 at 12:39:03 From: Jason Subject: Gallons/inch of height for a trapezoidal solid I have the same shape. Getting the volume is the easy part, but I need to know if I were to fill this shape with a liquid, how many gallons would there be per inch of height? I understand how to do this with a rectangular volume, but I can't get the vol/height ratio for this type of geometric solid. Thanks a lot!
Date: 08/27/2001 at 16:31:41 From: Doctor Peterson Subject: Re: gallons/inch of height for a trapezoidal solid Hi, Jason. If by "gallons per inch of height" you just mean the volume of a full container divided by its height, just use the formula (with dimensions in inches) to get cubic inches, divide by 231 to get gallons, and divide that by the height. But that doesn't mean much. For any container other than a cylinder, the ratio will vary with depth, and I think you want to vary the depth. I'm going to assume that you mean you are partially filling the container, and want to know the volume at different levels. That is, you want the volume as a function of height. To review the situation, here's the picture: a2 +---------------+ b2/ / \--------- +---------------+ \ | / | \ | /.................|.... + | h / | / | / | / | / | /b1 ------ / | / / |/ +-----------------------+ a1 V = [a1*b1 + a2*b2 + (a1*b2 + a2*b1)/2] * h/3 Now, the width in the "a" direction changes linearly from a1 to a2, making it a = a1 + (a2-a1)k = (1-k)a1 + ka2 at kh units up from the bottom (where k varies from 0 to 1); likewise, b = b1 + (b2-b1)k = (1-k)b1 + kb2 The volume of the solid with height kh, base a1 by b1 and top a by b is V(k) = [a1*b1 + a*b + a1*b/2 + a*b1/2] * kh/3 = [a1*b1 + ((1-k)a1 + ka2)((1-k)b1 + kb2) + a1((1-k)b1 + kb2))/2 + ((1-k)a1 + ka2)b1/2] * kh/3 = [a1*b1 + (1-k)^2 a1*b1 + (1-k)ka1*b2 + (1-k)ka2*b1 + k^2 a2*b2 + (1-k)a1*b1/2 + ka1*b2/2 + (1-k)a1*b1/2 + ka2*b1/2] * kh/3 = [(1 + (1-k) + (1-k)^2)a1*b1 + ((1-k)k + k/2)(a1*b2 + a2*b1) + k^2a2*b2] * kh/3 = [(3-3k+k^2)a1*b1 + k^2a2*b2 + (3k-2k^2)(a1*b2 + a2*b1)/2] * kh/3 If we use the actual liquid level L instead of the ratio k, we can replace k = L/h and get V(L) = [(3h^2-3Lh+L^2)a1*b1 + L^2a2*b2 + (3Lh-2L^2)(a1*b2 + a2*b1)/2] * L/(3h^2) This gives volume as a function of depth. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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