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First of all, Harper gave a great answer to a similar question here, but I'm having trouble making it work for an irregular shaped roof with an angle that is not 45 degrees.

My roof contains an internal drain, with the top right and bottom left sides sloping towards it.

I have attempted to make the slope myself following the reasoning from the above post but I'm not confident it's correct. I've stuck to a slope of 2:100 and tried to use the Pythagorean theorem to find the hypotenuse of the angle which is 65 degrees.

Is what I've done correct?

sketch of flat roof

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    As impressive as Harper's answer is, in the field no carpenter is going to do this math. He's going to pull drylines between critical points and take actual measurements. You're dealing with an existing roof, which isn't perfectly flat itself, so the math may not apply well anyway. If you try doing it on paper you're bound to be making at least some of the cuts twice after you find that things don't fit well. – isherwood Oct 4 '19 at 20:57
  • I'd agree that no tradesman out on a job like this would do all this math upfront. I like the idea of using math in this way -- it might generally be overkill -- but it's a fun DIY project for me. You mentioned taking measurements at critical points, how would you go about getting the slope right on a roof like this? – JamesDean Oct 4 '19 at 21:41
  • Establish a level line however you like and measure the difference at each end. That's the very definition of slope. – isherwood Oct 7 '19 at 12:41
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The slope of the roof is 2/100 so the elevation of the roof at and given distance from the drain can be expressed proportionally 2/100=elevation/distance or e=(2d)/100. So yes, 236 from the drain the elevation is (2*236)/100 = 4.72. If you can measure with a tape or step off the distance with dividers you can dispense with any more math than that.

Basically on the building you can measure the distance from the drain to any point on the roof and mark the height. Or if you're trying to cut strips you can find the start and end heights as above then snap a chalk line between them.

As a practical matter you'd probably want to increase the slope slightly beyond code and pick a few critical points such as corners and the midpoint of long edges to set the height. The reason being that you probably a) can't build a perfect cone with construction lumber and b) the rest of the building probably isn't perfect either. Giving yourself a margin for error means that every point in between should have at least the minimum slope.

By the way, the angle is irrelevant to the Pythagorean theorem which states the length of the hypotenuse is equal to the square root of the sum of the squares of the legs.

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Start by laying out your valley as you have done, install the valley boards (you'll also get a valley at approximately 90 degrees to this valley, where the planes of the two ends of the house meet, so leave space for that)

Now all you need to do is construct the planes so that they slope at an angle approx 45 degrees to the valley in the general direction of the drain 1:100 is probably enough slope for the planes.

you also want the slope to be away-from or parallel with the edges of the roof to prevent rivers from forming at the edges of the roof, anything between (30 degrees to 60 degrees direction of slope relative to a 2:100 the gutter will get you at-least 1:100 slope on the plane)

Making the slope go one way makes the lines of equal height (isoheights) go perpendicular, so you can copy the height of the valley board along along the lines of equal height. This short-cut can be seen in Harper's answer, although he doesn't call this result out, and in-fact disguises it by using two colours

The extra valleys caused where the planes meet will be where both planes have the same height. the direction these take can be determined by projecting lines of equal across the two planes and seeing where they intersect.

The bottom left corner is problematic, an angle of less than 30 degrees to the valley is needed there to avoid making a river against the edge of the roof. that's going to result in too little slope, you may need to steepen the valles in that direction to compensate.

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Thanks Mathew Gauthier and Jasen. I used a combination of your answers to give something I feel confident about doing.

From your answer Mathew, I'll be using the formulae you gave to easily workout the distance from the drain to the joists which intersect on the red line (the roof joists run horizontally) where I can then use the shortcut that you clarified, Jansen, about isoheights.

I decided to keep my slope at 45 degrees to simplify all the calculations and make it easy for me to make the firring strips for the joists, even if it'll mean that I'll have to trim a few joists and sister them with smaller ones. enter image description here

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