One right triangle, three answers
The support height is the opposite side, the set-back is the adjacent side and the ladder is the hypotenuse. Set-back is the height divided by the tangent of the angle, rail length is the height divided by the sine. That is the whole model.
The defaults: a support twenty feet up at 75.5 degrees puts the feet 5.17 feet out and needs 20.66 feet of rail to get there. Add three feet of rail projecting past the landing and the ladder is 23.66 feet from feet to top. A two-section extension ladder with three feet of overlap loses that three feet to engagement, so the sections have to add up to 26.66 feet — which is why the ladder for this job is a 28 footer and not a 24.
That gap is worth sitting with. The height is twenty feet, the ladder is twenty-eight, and every one of those eight feet has a reason: 0.66 for the angle, three for the projection past the landing, three for the overlap, and the last 1.34 because ladders come in four foot steps.
Why the angle matters more than it looks
Set-back changes gently with angle. At 65 degrees a twenty foot support wants the feet 9.33 feet out; at 75.5 it is 5.17; at 80 it is 3.53. Those are ordinary looking differences, a few feet on the ground.
The sideways push does not behave gently at all. Take 250 pounds of person and tools sitting three quarters of the way up. At 80 degrees the feet are being pushed sideways with about 33 pounds. At 75.5 it is about 48. At 70 it is 68, at 65 it is 87, and at 60 it is 108. Between the shallowest and steepest of those the ground has gone from a third of the picture to more than three times what it was, while the ladder still looks like a ladder leaning on a wall.
The reason is the cotangent. Near vertical it is small and changes slowly; the further you lay the ladder down the faster it climbs, and it is unbounded at horizontal. People calibrate on how the set-back looks, and the set-back is the thing that is changing least.
Overlap is length you paid for and cannot use
An extension ladder is sold by its nominal length, which is the sections added together. Extended, the sections have to stay engaged by an overlap that the manufacturer specifies and that gets longer as the ladder gets longer, so the usable length is always less than the nominal. Three feet on a two-section ladder, and on a three-section ladder you lose it twice.
Two consequences worth knowing before you go to the yard. First, sizing off the nominal number leaves you short by the overlap every time. Second, the projection past the landing comes out of the same budget — that is rail measured along the ladder, not vertical height, and at 75.5 degrees three feet of rail is only 2.9 feet of height.
What this page refuses to do
It does not tell you the angle to use. There is a figure on the label of your ladder and in the instructions that came with it, and that is the one that applies to that ladder. It does not tell you whether the ladder is rated for the weight, whether the feet will hold, whether the top is secured well enough, or whether a ladder is the right access for the job at all. The push figure is a statics result on an idealised ladder against a smooth vertical support, and it is offered because the way it changes with angle is genuinely counter-intuitive, not because it decides anything.
Where a ladder stops being the answer, the scaffold bay calculator covers the frame scaffold version of the same access problem and the boom lift reach calculator covers the machine version. For the angle expressed as a grade or a ratio, the slope and grade calculator.
Questions people ask
How far out should the base of a ladder be?
That is set by the angle the manufacturer specifies on the label of the ladder you are using, and the geometry follows from it: set-back is the support height divided by the tangent of that angle. A support twenty feet up at 75.5 degrees puts the feet 5.17 feet out; the commonly quoted one-out-for-four-up ratio is 75.96 degrees and gives five feet. Use the figure on your ladder rather than a number from a web page.
What length extension ladder do I need to reach a 20 ft roof?
Longer than you would guess. At 75.5 degrees you need 20.66 feet of rail just to touch the support, plus whatever projection past the landing the instructions call for, plus the overlap the sections lose to staying engaged. Three feet of each turns a twenty foot height into 26.66 feet of nominal ladder, which is a 28 footer once you round to the sizes actually sold.
Why does the ladder angle change the load on the feet so much?
Because the sideways push depends on the cotangent of the angle, and the cotangent climbs fast as the ladder is laid down. The same 250 pounds three quarters of the way up pushes the feet sideways with about 33 pounds at 80 degrees and about 108 pounds at 60. The set-back only changes from 3.5 to 11.5 feet over that range, so what you see changes far less than what is happening at the base.
Why do extension ladders never reach their stated length?
The sections have to stay overlapped when extended, and that overlap is length that exists but does not add reach. A two-section ladder loses it once, a three-section ladder twice, and the required overlap grows with ladder length. The nominal size on the label is the sections added together, not the length you can lean against something.
Does the projection past the landing count as vertical height?
No, and mixing the two is a common sizing error. Projection is measured along the rails. At 75.5 degrees three feet of rail is about 2.9 feet of vertical height, and at shallower angles the difference grows. When you are adding it to the length you need, add it as rail length, which is what this page does.