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Calling all Mathematicians and How insulated is a Dome

Framing the barn continues. We try to do at least 2 hours of work on it a day unless the weather is bad.

We have answered all the questions from zoning and the building department so our dome permit process is back on track and it is time to pay the fees. On the list of requirements we had to fill out a Res-Check for the building department. It is a very weird document to fill out if you are building a dome! It is so weird that I am not even sure it is relevant but I filled it out anyway. The hardest part to answer was related to the exterior. They want to know the surface area of all exterior walls and ceilings including what direction each faces. Ummmm......where is it wall and where is it ceiling? SO I just lumped it all together under one wall with no direction selected. This did not end my frustration. Calculating the surface area of the dome really threw my math challenged brain for a loop. I eventually kinda estimated in the end. Actually, if I am being honest, I really estimated! We will call it a somewhat educated guess.
I think I have a new wine I need to try! I definitely need it after trying to do the math!

Anyone out there that is really good at challenging math want to give it a shot?

Here is the floor plan and elevations again along with a photo of the interior of a similar dome.
It looks a bit like Mickey Mouse in the floor plan so I will use that to help you understand my math difficulties. If Mickey's ears are 16 feet high and 32' diameter domes and Mickey's face area is 18' high and made up of two overlapping 32' domes what is the total surface area? Oh yeah, I don't actually know how much they overlap....I just know that the total perimeter is 246 feet. If you look close in the floor plan you can see tiny + signs that indicate the center of each of the four 32' domes.

Are you stumped? I was! It is like a high school math nightmare to me! The different heights and the partial overlapping is just way to much for my math challenged brain.
Of course it may just be me. I do find myself scratching my head at times when my 5th grader has a hard math word problem for homework. I try to blame that on all the weird new ways they are teaching math now compared to the way I learned it. If I am honest, I have to admit that it is mostly my artsy brain. I got through the required college math credits with A LOT of help from one of my friends and a TA that would give practice quizzes that were EXACTLY the same as the exams with only the numbers changed.

You can cheat and go online to find dome surface area calculators BUT it won't help with the overlapped parts. It is all a bit silly anyway. The point of the Res-Check is to make sure that you are meeting the bare minimum required by the Department of Energy. I think we are exceeding by more than a little even if our Res-Check doesn't recognize it.

Check out this photo study that someone did using infrared (or purple in this case) cameras in -25 degree Celcius Canada of a dome workshop vs. well insulated traditionally built buildings next door. Super cool!
 This one is just a photo of the dome in question. It is located in Canada and at the time of the photos the temperature was -25 degrees C.

This image is the dome in the first picture. The shot is a bit more from the side and you can see the heat loss from around the insulated garage door.

This image is of the back side of the dome. You can see a little heat loss around some openings.

This is a photo of a building next door to the dome. It is considered well insulated by traditional standards.

This is also a building next door. It clearly isn't as well insulated.

Our dome passed the Res-Check but it says that we are only 30% more efficient than the bare minimum of requirements. This includes factoring in all the other energy efficient things we are putting in. I feel like the document doesn't represent what we are building very well at all but as long as it passes that is all that matters to the town and to the Department of Energy.


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