The ‘skin’ of a tipi is very nearly a semicircle. If it is a ‘normal’ 60 degree (cone) tipi, then (apart from a slight overlap for keeping out drafts), that’s exactly what it is. A semicircle with radius ‘R’ (eg 10’ – the width of the widest roll of Tyvek) will make a tipi which is approximately 10’ wide at the base. (C = 2xPixR – divided by 2 for a semicircle; D= C/Pi, so C = 2x3x10 = 60/2=30/3=10 – taking Pi as approx ‘3’). So, any width of fabric will make a tipi which has a diameter the same as its width.
You will need a piece that is twice as long as its width to make your tipi. If you cut out a little more of a circle than a semicircle, you will get a tipi which has a less acute angle than 60 degrees (say 45 degrees) and which is correspondingly wider (and shorter). (You would need to stick this piece on with Tyvek tape). You can work out how wide by dividing the piece’s circumferential length by 3. This is close enough.
A 10’ wide tipi is quite a handy size, (will obviously sleep at least four people) but is hard to erect unless you tie the ‘skin’ to the pole first (otherwise you can’t reach). If you are using only one pole you will need someone to hold the pole whilst you peg the sides down. That’s why folk normally used 3 or more poles (usually six) tied together at the top where you attached the skin’. You can even pitch it with no poles by throwing a rope over a branch.
You can cut the circular edge into eg six equal straight lines (a hexagon) if you like, and it will still pitch flush to the ground, and give better angles to the wind. Of course this works even better if you catenary cut them: http://www.theultralighthiker.com/catenary-curves/
You can make such a tipi out of a readily available blue poly tarp for less than $20 if you want (I have). If you are going to have a fire in your tipi, you must beware of carbon monoxide. You need airflow in at the bottom and out at the top. Any open fire is almost impossibly smoky. The Plains Indians must have been plagued by lung ailments. A chimney is a great idea, failing that some sort of chimnea arrangement. Titanium Goat (http://www.titaniumgoat.com/cstove.html) has lightweight (titanium) stoves and chimneys for just this purpose. The chimney will probably not be hot enough to melt the Tyvek at the top, but if you are worried you can wrap that section of pipe with some fibreglass cloth or etc.
As you will see at his website, he also makes really lovely tipi tents out of silnylon (pictured). This is something like what your Tyvek tipi will look like, (sans the cost!) Such an arrangement would suit car, motorcycle or horse based camping better than backpacking, but maybe you and your friends are quite strong!
New: Seamless Tyvek Tipi