Volume Calculations for Cylinder Shaped  Model Hot air Balloons

Homemade Plastic Bag Model Hot Air Balloons are generally shaped like pillowcases.  To calculate volumes, they can be compared with classic cylinders.  Here a balloon is imagined as a stack of circles, or ovals, with a top.  The volume is equal to the average area of the ovals, multiplied by an "Effective Height," which accounts for the material at the top of the balloon.

The simplest way to calculate areas for circles and ovals is to compare them with squares.  The perimeter is the most elegant geometric measurement.The quarter-perimeter though is usually more practical to use.

NOTE:  If the Perimeter of a circle, ie. its circumference, is divided by its diameter, the result is 3.1416..., or Pi.  So, the Perimeter / 2 Pi  equals the radius.  Similarly, 2 * Quarter-Perimeter / Pi  also equals the radius.

Area of a Square  =  Quarter-Perimeter ^ 2  (ie. the square of one of the sides)

Area of a Square  =  Perimeter ^2  / 16

Area of a Circle = Pi * Radius ^2  =  Pi * ( 2 * Quarter-Perimeter / Pi ) ^2  =  1.273 * Quarter-Perimeter ^ 2  ( ie. 4/Pi)

Area of a Circle  = Pi * ( Perimeter / 2 * Pi ) ^2  =  Perimeter ^2  / 12.566 ( ie. 4 * Pi  ) ( ie. 16 / 1.273)

So, with equal perimeters, circles have around 27% more area than squares.Moderate ovals have around 20% more area than squares, and wide ovals have around 15% more area.  So, ovals have around  5 - 10%  less area than circles.

What if the scale changes?  Twice the perimeter gets four times the area. Four times the perimeter gets sixteen times the area.  So the change in area equals the square of the change in perimeter.  Basically scale and measurement are the same thing.

Table for Approximating Balloon Volumes

The following table assumes that balloons are round.  To account for the material at the top of the balloon, and derive an "Effective Height," the presumed radius is subtracted from the "Material Height."  In reality, this overcompensates.  But it does offset for a small amount of tapering and ovalness.  It would be more accurate to subtract around 80% of the radius.

NOTE:  The table is set up to be mathematically consistent, not to correspond directly to actual balloon volumes.  To adjust for highly oblong balloons, volumes can be discounted by 5-10%.  To adjust for tapering, half-perimeters can be discounted.  So, in general, assume the calculations to over-approximate actual finished balloon volumes by up to several cubic feet.

Starting Approximations for Volume, Material Area and Weight for Different Sized Balloons

 Half-Perimeter 2 Feet Wide 3 Feet Wide 4 Feet Wide 5 Feet Wide 6 Feet Wide Quarter-Perimeter 1.0 feet 1.5 feet 2.0 feet 2.5 feet 3.0 feet Area of "Square" 1.00 Square Feet 2.25 4.00 6.25 9.00 Area of  "Circle" 1.27 Square Feet 2.86 5.09 7.96 11.46 Radius / Diameter .64 / 1.27  feet .95 / 1.91 1.27 / 2.55 1.59 / 3.18 1.91 /  3.82 4 1/2 Feet High 5 Cubic Feet * 10 Cubic Feet 16 Cubic Feet ** 23 Cubic Feet 29 Cubic Feet Material Area 18 Square Feet 27 Square Feet 36 Square Feet 45 Square Feet 54 Square Feet Weight -1/3 mil .40 oz. .60 oz. .80 oz 1.00 oz 1.20 oz. Weight - 1/2 mil .60 oz. .90 oz. 1.20 oz. 1.50 oz. 1.80 oz Area/Vol. ratio 3.60 ~ 2.70 ~ 2.25 ~ 1.96 ~ 1.86 ~ 6 Feet  High 7 Cubic Feet 14 Cubic Feet 24 Cubic Feet 35 Cubic Feet 47 Cubic Feet Material Area 24 Square Feet 36 Square Feet 48 Square Feet 60 Square Feet 72 Square Feet Weight - 1/2 mil .80 oz. 1.20 oz. 1.60 oz. 2.00 oz. 2.40 oz. Weight - 1/3 mil .53 oz. .80 oz. 1.07 oz. 1.33 oz. 1.60 oz. Area/Vol. ratio 3.43 ~ 2.57 ~ 2.00 ~ 1.71 ~ 1.53 ~

*Classic Dry Cleaner Bag Balloon  (Typically powered by around twenty birthday candles)
**  Standard Homemade Plastic Bag Balloon  (Typically powered by around forty birthday candles)

As described, a balloon's "Effective Height" is approximated by subtracting its presumed radius from its "Material Height."  Since more width increases the radius, the different balloons are not really in scale with each other.

Blimp shaped balloons can be calculated as either tipped over cylinders, or as short fat cylinders.Either way the volume calculations are fairly close to each other.  In both cases, to account for the extra ovalness and tapering, the volume should be discounted more than for rounder balloons.

To summarize, if a balloon is made wider, and its effective height stays the same, the volume changes by the square of the increased width.  If a balloon is made taller, and the width stays the same, the volume change is the same as the increased height.

NOTE:  The Area to Volume Ratio is used to approximate a balloon's weight per cubic foot, based on its material weight.  For lightweight Plastic Balloons, this is usually not much of an issue.  For heavier Paper Balloons though, this ratio is very important in evaluating a balloon's overall aerostatic potential.

## Geometry of Scaling -- For Dimension, Volume and Surface Area

Cylinder shaped balloons are in scale with each other if the relative changes in the height and the width are the same.  The following formulas can be used to compare scaled balloons, or any other scaled three-dimensional objects:

Change in Volume =  Change in Dimensions ^ 3
Change in Surface Area = Change in Dimensions ^ 2

Change in Volume =  Change in Surface Area  ^ 1/2  ^  3  =  Change in Surface Area  ^  3/2
Change in Surface Area  =  Change in Volume  ^ 1/3  ^  2  =  Change in Volume  ^  2/3

NOTE:  Mathematically, "Powers are Powered" by multiplying the exponents times each other.

The formulas can be demonstrated, by alternately doubling the dimensions, the surface area and the volume:

Base Two Geometric Sequence and Exponent Table -- Useful for a Range of Balloon Calculations

 Power Zero 1/3 1/2 2/3 One 3/2 Square Cube Four Five Six BaseTwo 1 1.26 1.414 1.587 2 2.828 4 8 16 32 64

Here are the results:

Doubling the dimensions increases the surface area by four times (ie. 2^2) and increases the volume by eight times (ie. 2^3).

Doubling the surface area increases the dimensions by 1.414 times (ie. 2^1/2) and increases the volume by 2.828 times (ie. 2^1/2^3 or 2^3/2).

Doubling the volume increases the dimensions by 1.26 times ( ie. 2^1/3) and increases the surface area by 1.587 times
(ie. 2^1/3^2 or 2^2/3).

Summary:  A range of changes in dimensions, area and volume can be calculated from the following table:

 Change^x Two Three Four Five Six Seven Eight Nine Ten ^ 1/3 1.260 1.442 1.587 1.710 1.817 1.913 2 2.080 2.154 ^ 1/2 1.414 1.732 2 2.236 2.449 2.646 2.828 3 3.162 ^ 2/3 1.587 2.080 2.520 2.924 3.302 3.659 4 4.326 4.641 ^ 3/2 2.828 5.196 8 11.180 14.670 18.520 22.627 27 31.622 ^ 2 4 9 16 25 36 49 64 81 100 ^ 3 8 27 64 125 216 343 512 729 10

Appendix:  Notes on the Base Two Exponent Table and its applications:

As described "Powers are Powered" by multiplying the exponents times each other.  For balloon mathematics this appears to be the only application.As examples, to square a number, the exponent is doubled.  To cube a number, the exponent is tripled.  Equivilently, the square root gets 1/2 the exponent, and the cube root gets 1/3 the exponent, etc.  Try it.  There are many examples on the exponent table.  One example is that 4^3/2 = 8.  Another is that 1.682^2 = 2.828

Also, numbers can be multiplied times each other by adding their exponents.  They can be divided by subtracting their exponents.  For balloon mathematics this application is not really used.  But it is important to know anyway, to understand how exponents work.  As examples, 1.189 times1.682 and 1.26 times 1.587 are both equal to one.

Other numbers besides two can also be used as the base for a geometric sequence, including those on the Two Scale.  As example, if four is used, each exponent gets half its previous power.  If the square root of two is used, ie. 1.414, then each exponent gets double its previous power, etc.  Hence scales can be shrunk or enlarged and still be compared to the Two Scale.

Alternately, any other number, definite or indefinite, except for zero, can be set up as the base, with the changes calculated relative to it.This can be done simply by declaring that a certain value is now to be considered as equivilent to one "unit."

NOTES ON TERMINOLOGY:

1)  Saying 2^1/2 (power) is the same as saying the square root of two.  Saying 2^1/3 (power) is the same as saying the cube root of two, etc.  Technically the term "root"  refers to the specific number which when multiplied by itself a certain number of times yields the starting number.  Hence saying the half root of two would technically seem to mean 2^2 (power).  Hence, the term "root" is avoided here, except as a shorthand way of referring to the 1/2 and 1/3 powers.

2)  As a general concept, or in groups, the term "power" is usually called an "exponent." Similarly, a "powered function" is usually referred to as an "exponential equation."  This wording convention is probably used because using the word "power" outside of the specific context of raising a number to a power might tend to get confusing.

NOTES ON SCALING CONCEPTS:

Sometimes model designers use a Base Three Scale, ie. 3, 9, 27, 81, etc.  From here combinations of the two and three scales can be used to create a Double Scale.  Hence, using a Single or Double Scale System the simplest scales are:

( 1.414 )23468912  16  18242732  36  48  54  64  7281  96  108  128  144  ...

People who build models might recognize some of these numbers.  On a practical level though, only the smaller scales would actually be used.  But why bother to use numbers along the so-called alternate scales sequence?.  Basically because the calculations tend to work out more evenly, and comparisons are easier to make, so the models are simpler to design, test and manufacture.

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