**If you feel able to do these projects... the Mark 3 and 4 will give you enormous satisfactions!**

In this guide I will explain you, with step by step procedures how to build these two balloon models. I decided to offer together the two models, because their design is similar, and the difference lies in some mathematical formulas that we will use.

A clarification on the form, the Mark III has the upper hemispherical while the lower part as a common hot-air balloons.This makes it more simple to design and build but compared to the Mark IV is slightly less stable. This does not mean that it is not a great balloon, always better than Mark I & II.

Instead, the Mark IV has in effect the shape of the common hot-air balloons and I guarantee you it was not easy to get to develop the formulas necessary to its development.

For don't write a mathematics book, I will avoid to describe the logic behind the calculations, but I will give you a step by step guide to follow, so everyone can try their hand at construction.

If it seems to you some data missing, I will explain what is and how to calculate it, and if i will not do it, it's because we have already done and then you just have to go back to see him again.

Well..... Let's start 😀

First of all we must know how much we want to do big the balloon, and consequently we can then calculate the rest.

Now for example I will design a balloon that will have a volume of 8 cubic meters, but of course you can change everything at will for it to be suitable to your needs, knowing that to lift a kilogram takes about 3 cubic m.

The part that follows, might initially seem complex, but simply follow the step by step instructions to complete the section about calculations in an easy and fast way.

We begin first thing to calculate the data of the sphere which has a volume (determined by me, but you can change it at will) equal to 8 cubic m using the following formulas. To help you understand the formulas and where to place the data obtained on the balloon being built, I added drawings and photos 🙂

In the case of double formulas between Mark III and IV, for convenience I will report only in the column of the Mark III.

#### First choose the model that you want to build.

**Mark 3**

We begin first thing to calculate the data of the sphere which has a volume (determined by me, but you can change it at will) equal to 8 cubic m using the following formulas.

**Mark 4**

First we decide to how many cubic m want to build the balloon, -in this case will be 8- and then we apply the following formula:

Cubes m to be used in the calculations = ( cubic m /5) * 6 = (8/5)*6 = 9,6 cubic m

Now that we have the new value of the volume, we calculate with the following formula the value of r.

Thanks to this formula we will calculate the r value.

The only information that we will enter into the formula is V, which is the volume found previously.

**r = 1,24 m**

Then we calculate the diameter:

**D = r * 2 = 2,48 m**

Now the circumference:

**C = D * π = 7,79 m**

Mouth balloon diameter:

**Db = D/4 = 60 cm (data variable according to requirements of design / comfort of use)**

Balloon mouth circumference :

**Cb = Db * π = 1,88 m**

**r = 1,31 m**

Diameter:

**D = 2,63 m**

Circumference:

**C = 8,28m**

Mouth diameter:

**Db = 0,65 m**

Mouth circumference:

**Cb = 2,06 m**

Now we decide the n° of gores which we have to do, they are vertical stripes that start from the mouth of the balloon and go up to the summit. Plus the number of gores will be higher and we get closer to the spherical shape and the desired volume -but we will also have more work to do, and whether it will be down instead of a ball we get closer to other geometric shapes (four gores = Square, 5 gores = pentagon, etc.)

For a good shape we can choose between 10 and 20 but there are no fixed parameters, 14/16 in doubt go well, suggest that they are equal, I'll use it 15 because I'm using a project that i had done long ago,but experience has taught me that an equal number is better, especially if your model is very big, this is because the junction points between the gores, we're going to attack the reinforced tape that holds the fuel and a possible payload (eg a camera).

**Gores N° = Sp = 15**

For construction convenience -given that the gores will be very long and cumbersome- we will design them divided into upper half and lower half.

We start now to calculate the size of the semi-gores which will make up the balloon:

Upper gore height:

**hss = C/4 = 1,94 m**

Low Side upper gore:

**Lb = C/Sp = 51,9 cm**

Angle at which to end, the upper gore:

**360/Sp = 24°**

Now we must decide how many "sections" will build the balloon, and we'll call these sections -point-

The higher the number of points selected and the form will be approximated to a sphere -points, like the segments is recommended to be equal- (recommended at least 8)

**Point number = Np = 8**

(In the picture below, for simplicity I have represented only 3. We'll use 8 in order to obtain a less edgy).

With the following formula, we calculate now h values, which we will use later. To find them we will have to replace the last value on the right in the formula with the corresponding value of h we want to find, for example, if we want to find h4, we will use the value -4, if we want to find h5, we will use the value -5 ecc...

h1 = (1,24/8) * (8-1) = 1,08

h2 = (1,24/8) * (8-2) = 0,93

h3 = 0,77

h4 = 0,62

h5 = 0,46

h6 = 0,31

h7 = 0,15

h1 = (1,31/8) * (8-1) = 1,14

h2 = (1,31/8) * (8-2) = 0,98

h3 = 0,82

h4 = 0,65

h5 = 0,49

h6 = 0,33

h7 = 0,16

For the point h8, we will do so:

**h8 = (r/Np)/2**

Now we will calculate a final point not mentioned before, with this other formula

**htop = [r/(Np*2)]/2**

h8 = 0,08

htop = 0,04

h8 = 0,08

htop = 0,04

Now that we have all the points h, we will use them to find the points A

Obviously A is calculated for a number of times equal to the number of points (Np), each time varying the value of h.

As before, if the formula we will use h1 result will be the A1 value, if we use h4 the result will be A4, and so on.

A1 = 1,22

A2 = 1,2

A3 = 1,15

A4 = 1,07

A5 = 0,96

A6 = 0,82

A7 = 0,59

A8 = 0,44

Atop = 0,31

A1 = 1,30

A2 = 1,26

A3 = 1,21

A4 = 1,13

A5 = 1,02

A6 = 0,88

A7 = 0,63

A8 = 0,45

Atop = 0,32

Various points A1, A2, A3..ecc.. we will need it to calculate the values L1, L2, L3 etc ..

The latter are very important because we will use them in the construction of the balloon, in fact indicate the gore width at different heights.

**L1 = ((A1*2)*π) / Sp**

**L2 = ((A2*2)*π) / Sp**

Along with the widths L, we calculate the h2.n values that will help us later.

Below, you will find in the two columns to the calculations for the values L and h2.n for balloon Mark III, and below them those related to the Mark IV

Beware to h2.8 and h2.top values, and do not panic if the results will be a negative number. It's normal. We will turn it into positive and we will use it.

Example: h2.8 = -125 → 125

**Mark 3**

L1 = 51,5 cm

L2 = 50,3 cm

L3 = 48,2 cm

L4 = 45 cm

L5 = 40,5 cm

L6 = 34,6 cm

L7 = 25,1 cm

L8 = 18,1 cm

LTOP = 12,9 cm

h2.1 = (r/Np)*1 = 15,5 cm

h2.2 = (r/Np)*2 = 31 cm

h2.3 = 46,5 cm

h2.4 = 62 cm

h2.5 = 77,5 cm

h2.6 = 93 cm

h2.7 = 108,5 cm

h2.8 = [(r/(Np*2))-r] =116,3 cm (apply only fol the latest point h2.n )

h2.top =([(r/(Np*2)) / 2] - r) = 120,1 cm (apply only for L TOP )

**Mark 4**

L1 = 54,4 cm

L2 = 52,7 cm

L3 = 50,7 cm

L4 = 47,3 cm

L5 = 42,7 cm

L6 = 36,8 cm

L7 = 26,8 cm

L8 = 18,8 cm

LTOP = 13,4 cm

h2.1 = (r/Np)*1 = 16,4 cm

h2.2 = (r/Np)*2 = 32.7 cm

h2.3 = 49,1 cm

h2.4 = 65,5 cm

h2.5 = 81,8 cm

h2.6 = 98,2 cm

h2.7 = 114,62 cm

h2.8 = [(r/(Np*2))- r] =122,8 cm (apply only for latest point h2.n )

h2.top =([(r/(Np*2)) / 2] - r) = 126,9 cm (apply only for L TOP )

**Come on, one last effort 🙂**

Now that we know the widths (L) of the various points, we have to calculate how far away from the base of the gore (H) they must be.

Once we have done we will have all the data to build the upper gores that will serve us for our balloon.

For find the H heights we will use two different formulas depending on the model that we are planning.

Obviously each point h2.n used in the formula, will generate at the end of the same one point Hn (if for example we will use H2.4 as a result we have the point H4)

Attention:

In the Mark IV, the H points found indicate the height of each segment, so to know the total height from the base of the gore to the point where to place the corresponding L, will be added the previous H point.

Example :

H3 = Formula result + H2+H1

H4 = Formula result + H3 + H2+H1

**Mark III**

**Mark IV**

H1 = [(sin^-1(15,5/124))/90] * 194 = 15,5 cm

H2 = 31,3 cm

H3 = 47,7 cm

H4 = 64,9 cm

H5 = 83,7 cm

H6 = 105,1 cm

H7 = 132,0 cm

H8 = 150,8 cm

HTOP = 163,6 cm

H1 = 10,98 cm

H2 = 11,57 → + H1 = 22.55 cm

H3 = 12,02 → + H2 + H1 = 34,57 cm

H4 = 13,54 → + H3 +H2 + H1 = 48,11 cm

H5 = 15.46 → + H4 +.... = 63,57 cm

H6 = 17,76 → + H5 +.... = 81,33 cm

H7 = 27,29 → + H6 +.... = 108,62 cm

H8 = 18,8 → + H7 +.... = 127,42 cm

Htop = 13,28 → + H8 +.... = 140,7 cm

We calculate now the lower gore data.

Ll = Large side = Lower side upper gore = 51,9 cm

The large side of the lower gore, of course will be equal to the low side of the upper gore as they have to fit together and then later be joined to form a complete gore.

The lower gore will only H and L points from 1 to Np / 2 and just.

**Np/2 = 8/2 = 4**

So we will use the values L and H up to L4 and H4

Note: For the balloon Mark III points H and L in question will be the same as those of the upper gore (then just copy them). The H points will be counted , however, starting from the top of the lower gore part, to fall. Instead for Mark IV Only points L will be equal, the H point we will have to recalculate with another formula.

**Mark 3**

**Mark 4**

H1 = 15,5 cm

H2 = 31,3 cm

H3 = 47,7 cm

H4 = 64,9 cm

L1 = 51,5 cm

L2 = 50,3 cm

L3 = 48,2 cm

L4 = 45 cm

TO RECALCULATE

LOOK DOWN

L1 = 54,4 cm

L2 = 52,7 cm

L3 = 50,7 cm

L4 = 47,3 cm

For calculate the lower gore points H of the Mark IV model we will use the following formula, and to results we will add the first value obtained from the previous H points in order to have the total distance from the base of the lower gore.

These points H will relate ONLY to the lower gore. Careful not to confuse them !!

H1 = √[(131-130)^2 + (16,4)^2] = 16,43 cm

H2 = 16,78 + H1 = 33,21 cm

H3 = 17,14 + H2 + H1 = 50,35 cm

H4 = 18,24 + H3 +.... = 68,59 cm

With the simple formula below calculate Lc (Short Side lower gore):

** Cb/Sp = 12,5 cm**

The height of the lower gore counted from the last point H (which would H_Np/2, so in our case H4) to its base must be obtained by the formula below:

It seems complicated, but we already have all the data to resolve it.

Mark 3

Hsi = 146,2 cm

Mark 4

Hsi = 153,71 cm

Now to find the total height of the lower gore we will add to the value just found the value H_Np / 2

Then add in our case the H4 value.

H4+146,2 = 64,9+146,2 = 211,1 cm

H4+153,71 = 68,59+146,2 = 214,79 cm

**Now-finally-the calculations are finished and we have all the data we need to build the envelope 🙂**