Floor radius formula of 3V 5/8 Icosa dome

Geodesic math and how to use it, Hugh Kenner, p102 – p103
angle x = 180 – theta
angle y = 180 – 90 – x
= 180 – 90 – (180 – theta)
= theta – 90
p = sin(y)
= sin(theta – 90)
= – cos(theta)

zenith altitude = r + p
= r – cos(theta)

floor radius = cos(y)
= cos(theta – 90)
= sin(theta)

altitude = r – cos(theta) [Eq. 14.3]
floor radius = r x sin(theta) [Eq. 14.4]

The author always leave behind the real answer as usual. I suspect the author ever built a real dome by himself all due respect. Because the book contains not a single dome photo or full dome data. It is considered a bible for Geodesic Dome mathematics but lack of practical example and obscure explanation of major formulas make the reader how to follow.

What I need to know is value of theta for getting floor radius. But the author again doesn’t explain how to find theta here.

Below is how I calculated the theta to get the floor radius on frequency three 5/8 icosahedron dome.
The image is generated by Geodome. It divides radius 9 even segment and uses 6 of them to construct dome.

The middle point of vertical line, diameter of the dome, is center of the circle and radius is half of the diameter. I assumed the length of diameter as 9 units. Half of 9 is 4.5.
I could make a right angle triangle whose hypotenuse is radius of the circle. The base of triangle is floor radius of the dome. The height is 1.5 units.

Apply Pythagoras theorem to find the base,
4.5^2 = 1.5^2 + x^2
x^2 = (20.25 – 2.25)
x = 4.2426

Apply arcsin or arccos to find the angle Phi,
Phi = arcsin(1.5/4.5) gives 19.471 degrees.

Another method is to simple variable manipulation of r.
Let r represents hypotenuse of the right angle triangle and radius of the circle.
Then the height of the triangle becomes one third of r. The radius is evenly divided by 9 segments.
Apply sine angle formula on Phi,
sin(Phi) = (1/3) x r / r
= 1/3
Phi = arcsin(1/3)
= 19.471 degrees

I can use the angle of Phi for any size of 3v 5/9 icosa dome generated by Geodome to find floor raidus.
And it makes me calculate the radius of dome with a floor radius. Because floor radius is dictates usable floor area of real size dome.

Floor area = Pi x (floor radius)^2

There are two examples.
When the floor radius, x is 20 cm, applying simple cosine angle law gives radius of dome.
20 = r x cos(Phi)
r = 20 / cos(phi)
= 20 / cos(19.471)
= 20 / 0.9428
= 21.21 (cm)

If the floor radius is 200 cm, the dome radius is 212.13 cm.
r = 200 / 0.9428
= 212.13 cm

I set radius of the real size dome as 213 cm, which I will start cutting lumbers soon.

About janpenguin

Email: k2.mountain [at] gmail [dot] com Every content on the blog is made by Free and Open Source Software in GNU/Linux.
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