A lean-to roof, also known as a shed roof, is the simplest roofing design and the simplest to build because the roofing material usually covers only a single rectangular plane. You can simplify the construction of your lean-to roof even more by building your own trusses. Perhaps the biggest advantage of that is that you can mass produce them on the ground before installing them. Under most circumstances, it takes only a day to install trusses, so the interior of the structure is exposed to the elements for only a short amount of time.
Find the total rise of the roof using the formula r = ps/12, where r is the rise, p is the pitch and s is the span. For example, if the pitch is 4, and the span is 9 feet 6 inches, r = 4*9.5/12 = 3.167 feet. Converting that to feet and inches, the rise is 3 feet 2 inches. If you know the rise and want to find the pitch, the formula is p = 12r/s.
Calculate the length of the top chord with the formula h = sqrt(r^2 + s^2), where h is the length of the top chord, measuring from either side of the chord from plumb cut to plumb cut. For example, h = sqrt(3.167^2 + 9.5^2) = 10.014 feet. Converting that to feet and inches, the length of the top chord is 10 feet 3/16 inch. Use the common-rafter pitch value of 4 on a speed square to lay out the plumb cuts.
Lay out the king post whose length to the long point is the rise minus the width of the framing material. For example, if you're using two-by-fours, the length of the king post would be 3 feet 2 inches minus 3 1/2 inches, which is 2 feet 10 1/2 inches to the long point. This cut can also be laid out using the common-rafter pitch value of 4 on a speed square.
Lay out the bottom chord whose length to the long point is the same as the span. Since the angle on the bottom chord is very acute, it's easiest to use the edge of the top chord as a guide for drawing the angle after arranging the pieces as they will be assembled. The three pieces will form a right triangle with the ends of the king post butting the edges of the top and bottom chords. In other words, sandwich the king post between the top and bottom chords. Lay the three pieces in this position on the floor with the bottom chord nearest to you. Since the bottom chord is still uncut, lay the top chord over the bottom chord, lining up the top chord's nearest edge to the bottom chord's long point. Run a pencil along the edge of the top chord to mark the angle on the bottom chord.
Cut and assemble the top and bottom chords and the king post.
Lay out the vertical webbing members such that their on-centre distance is 1/3 of the bottom chord's short side: d = (s - w/p)/3, where d is the on-centre distance, and w is the width of the framing material. For example, d = (9.5 - 3.5/4)/3 = 2.875 feet. Converting that to feet and inches, the on-centre distance is 2 feet 10 1/2 inches. Position the truss on the floor with the bottom chord near you and the king post on the right. Hook your tape measure on the king post to make marks at 2 feet 10 1/2 inches and 5 feet 9 inches on both the top and bottom chords. Be sure the tape measure is parallel to the bottom chord when measuring. You will use these marks to align the right edges of the vertical webbing members.
Mark and cut the vertical webbing members. The length of the longer of the two is v1 = (p(s - d) - w)/12, and that of the shorter is v2 = (p(s - 2d) - w)/12. These lengths are to the long point, and the angle is the same as the king post's angle. For example, v1 = (4_(9.5 - 2.875) - 3.5)/12 = 1.917 feet. Converting that, the first vertical member's height is 1 foot 11 inches. This piece should line up with the marks made in Step 6 at 2 feet 10 1/2 inches. For the second, v2 = (4_(9.5 - 2*2.875) - 3.5)/12 = 0.958 feet, which is 11 1/2 inches. Line up this piece with the marks at 5 feet 9 inches.
Lay out the diagonal webbing members. Rather than trying to calculate lengths and angles, it's easier to assemble the truss with the vertical webbing members and use the truss itself for marking the diagonal pieces. Place a two-by-four under the truss so that its centre line passes under the point where the right edge of the longer vertical webbing member meets the top chord. At the other end, the centre line should pass under where the left edge of the king post meets the bottom chord. With the two-by-four in place, mark where the cuts need to be made. Lay out the other diagonal member in the same way, but it's centre line will extend from the top of the shorter vertical member to the bottom of the longer one. Once the diagonal members are laid out and cut, use them as prototypes for laying out all of the other trusses.
The webbing design may vary according to the circumstances. Use more or fewer webbing members, depending on expected loads and the truss's span. Use truss plates or gussets for securing all the joints.
Have your roof design and trusses properly engineered. Adhere to all local building codes.