Making a Cone from a Flat Sheet

I wanted to make a transition fitting for my dust collector, that would go from 10" diameter at the top to 4" diameter at the bottom - basically a big funnel to drop sawdust into a bucket.  I remembered from pre-school that you can take a circle of paper, cut a wedge out of it, and it will make a cone.  Cut the bottom off the cone, and there you go.  Yeah, not so easy if you actually want to make a shape with predefined dimensions.  After a few days of head scratching, I figured out some fairly simple formulas to cut a cone of whatever size I want out of paper, plastic, or sheet metal.  I'll go through my so-called thought process, but if you just want the formulas, scroll to the bottom now.

I start with the top radius (Rtop), bottom radius (Rbot), and height (H) of the desired cone.

Start with these three dimensions

The first thing I do is figure out the angle of the cone, given all three measurements.  This was the key realization for me, because it's easy to make random cones that fit either the top OR the bottom, but the slope between the two sizes determines the entire layout.  Here's a cross section of half the cone.  We can draw a little right triangle with top dimension Rtop - Rbot, and height H.  The tangent of the angle (depicted by the theta symbol) is the opposite side length divided by adjacent side length (remember 10th grade?) so the angle can be determined by taking the inverse tangent of (Rtop-Rbot) / H.

Figure out the Cone's Angle

Now extend the bottom and center lines to where they meet below the cone.  This point is the center of the cutting pattern.  The theta angle is the same down here, and can be used to figure out the distances from the center to r1 and r2, the radii of the two arcs to be cut.  Again go back to basic trig and remember that the sine of an angle in a right triangle is the opposite side length over the hypotenuse length.  Sin theta = Rbot / r1, and the same for the larger triangle.  Rearrangement gives r1 = Rbot / sin theta and r2 = Rtop / sin theta.

Calculating the radii of the cuts

Now you have two radii and you can cut a couple of concentric circles.  The last part we need is the angle of the wedge we need to make that flat circle into a cone.  Start with the radius of the cone's top.  That gives the circumference = 2 x pi x Rtop.  So that's the length of the arc we need.  If we took our larger cutting radius (r2) and cut a full circle, it would be 2 x pi x r2 in length.  The ratio between those lengths is the amount of the full circle we need, but 2 x pi x Rtop / 2 x pi x r2 can be simplified to simply Rtop / r2, which will give the ratio we need.  You could also do this with the lower Rbot and r1 dimensions, which is a good check because it should be the same.  Multiplying that ratio by 360 degrees (in a full circle) gives the angle we need for our cutting pattern.

Determining the angle for the cut pattern

You will likely need to leave some material to overlap where the seam of the cone joins, for gluing / soldering / duct tape / whatever, and also some extra around the top and bottom to mate the cone to other things, so I recommend creating a scale mockup out of paper before you commit to anything big or expensive.

Here are the formulas for making a pattern for a funnel or cone out of flat stock.  Refer to the figures above for reference.

Inputs:
Rtop: radius of the top (wider) opening
Rbot: radius of the bottom opening
H: vertical height through centerline

first calculate theta: the cone's half-angle = tan-1((Rtop-Rbot) / H)

Outputs:
r1: radius of inner circle to cut = Rbot / sin(theta)
r2: radius of outer circle to cut = Rtop / sin(theta)
A: angle of arc = (Rtop / r2) x 360

Automatic irrigation

I don't like to waste my time dragging a hose or watering can around the yard.  Our summers are very dry, and irrigation is a must, so automating that is one of my priorities.

I've tried the little battery powered ball valves that go on a garden hose connection.  They're garbage, and they're expensive.  The mechanical parts are weak, the batteries die and / or corrode, etc.

A much better alternative is a "real" lawn sprinkler valve.  They operate via a low-voltage signal, that controls a little solenoid, which in turn influences the distribution of pressure inside the valve, moving a big rubber diaphragm back and forth to control the water flow.  It sounds complicated, but they're very reliable, can move a lot of water, and take a certain amount of abuse (like freezing).

Rainbird, Orbit, Hunter, and Toro all make fine valves.  You can find them at Home Depot or the like, or Amazon.  The standard signal is 24V AC, which you can supply in a couple of ways.  The easiest is to just buy a sprinkler timer.  A cheap one is about $20 and will control more than one valve.  Another option is to  use some other timer, and use it to control a 24V power supply.  A cheap Intermatic  24h timer and a "wall wart" power supply from your scrap pile will work fine.

There are other control methods, too.  When I was a kid, I fitted a misting system into my grandfather's small greenhouse.  Instead of a timer, I took a mercury switch from an old thermostat, and built a see-saw type mechanism from brass rod and pvc pipe.  One end of the see-saw had a cloth pad attached to it, and the other end had some weights.  The mercury switch was attached to the arm and closed when it tipped away from the cloth.  The contraption sat on the bench with the plants, and the idea was that whenever the pad dried out, the see-saw would tip, closing the mercury switch and providing power to the sprinkler valve.  It worked great, watering more on hot days and less on cool days and basically keeping the moisture levels very constant.

Review: Imperial Blades for the Dremel Multi-Max

I got a little variety pack of Imperial blades from Amazon, mainly to try the MM300 blade that's supposed to cut nails. I have the corded Dremel Multi-Max (model 6300). My first impression was how much thicker and heavier the blades are, compared to the stock Dremel blades. Seems like a good thing. Unfortunately, the wide MM150 blade is so heavy that it vibrates the tool like crazy, making it tough to control and uncomfortable to hold. I'm sure the bearings aren't liking it much either.  The cap screw that holds the blade on also vibrated loose after a few seconds.  I don't really think this blade is usable on this tool.

I did use the MM200 blade (1-1/4" coarse-tooth) to flush-trim a few feet of plywood that was sticking slightly out of a framing assembly in a skylight.  It worked really well, one of those situations where I would have been at a loss to do the job without this saw.  The blade itself was appropriate for the job, cutting quickly and surely.  I'm happy with the performance.  Unfortunately, the universal hole pattern didn't quite fit my Dremel saw, and I sort of had to pry it off the little pegs after removing the retaining bolt, which bent the blade a little.

The MM300 metal cutting blade did live up to my hopes.  It can reach into framing gaps and cut nails out in corners where you can't go with a reciprocating saw.  I don't think there's a better solution if you want to do clean, precise dissection of an old house.  I got in a weird little corner at the end of a kneewall and cut through the nails to remove a stud very easily, with no damage to anything.  I'll be able to reinstall the stud when I'm done.

I really want to like these blades.  They're made in the USA and they're a lot cheaper than the Dremel branded options, but I wouldn't bother with the MM150 for the Dremel.  It might work better on a heavier tool like the Fein Multimaster.

Quick Drywall Compound

Setting type drywall mud is really useful for quick repairs.  It's basically plaster of paris with fillers and retarders to lengthen the set time, and it can be sanded and ready for paint in 30 minutes.  I've found that a small plastic bag is really handy for mixing up a small batch with no mess.  Just put a half cup or so of powdered compound in a sandwich bag, add in some water, and squish it with your fingers to mix it until it's smooth.  Add a little more water if you need to.  Twist the bag shut, cut a corner off, and squeeze the compound where you need it.