Calculating the volume of a 4V kruschke 7/12 Dome

[John Morris]

Hi there,

Can anybody explain how to calculate the volume of a 4V kruschke 7/12 Dome?

I understand the formula V = ⁴⁄₃πr³ for calculating the volume of a sphere and then dividing that by 2 to get the volume of a hemisphere but I'm unsure about calculating 5/12, 7/12, 3/8, 5/8, 4/9, 5/9 etc dome volumes.

Any help would be appreciated.

Thanks

John

[Gerry in Quebec]

Hello John,

The volume of a 4v icosa, 7/12ths dome based on the Kruschke method is 2.8460 times the spherical radius cubed.

Finding the exact volume of a geodesic dome is a rather tedious exercise. You need to know all the chord factors, the spherical radius, and the distance between the truncation plane (the dome "floor") and the centre of the dome's circumsphere.

 
1) Find the volume of the irregular tetrahedron (a type of pyramid) whose vertices are the centre of the dome's circumsphere and the three corners of a face triangle. Multiply the result by the number of triangles of that type.
2) Repeat step 1 for each type of triangle making up the surface of the dome.
3) Add up all the volumes and call the sum V1.
4) Calculate the volume of the irregular pyramid formed by the dome's 20 footprint vertices and the centre of the dome's circumsphere. Call that V2. If the dome is a hemisphere, V2 will be zero. If the dome is more than a hemisphere (as in the 7/12 truncation), the dome's volume is V1 + V2. If the dome is less than a hemisphere -- say, 5/12 -- then the volume is V1 - V2.

 

If, for your particular purposes, you don't need a highly precise figure, just a good approximation, then you might want to use the Excel spreadsheet posted here:

https://groups.google.com/forum/#!searchin/geodesichelp/idealized/geodesichelp/PFwIgOOfXbE/eIU99GfoxsYJ

This calculator estimates the surface area and volume of a triangulated dome. The inputs are the number of triangles, the dome's height, and the radius of the base (i.e., floor radius). The 4v icosa, 7/12 Kruschke dome has 200 triangles. When the spherical radius is 1 unit, the dome's height is 1.27639 units and the radius of the base is 0.96104. Plugging these numbers into the calculator gives a volume estimate of 2.8408 cubic units, pretty close to the 2.8460 value resulting from the detailed calculation.

- Gerry in Quebec

[John Morris]

Hi Gerry,

I just wanted to say a huge thank you for this.

I had a feeling it wasn't going to be a simple calculation but a close approximation is all I needed and this is perfect.

Many thanks

John