Hip roofs slope to eaves on all sides, unlike gable roofs that slope from a centre peak on two sides. Hip roofs require cutting more complicated angles on rafters but may provide more internal space than gables. A hip roof usually will be added to the side of a house when there is an addition to the first floor of a two-story house. The hip roof will run from the end of the addition to the side of the house.
Design the hip roof. Measure the length and width of the addition and decide the pitch, the slope of the roof. That might be a 6/12, a 6-inch rise for every foot of rafter run. Make two rafters to that pitch, using a framing square. Put the corner of the framing square on the bottom of a rafter board with the 6-inch mark on the short side of the square at the top of the board and the 12-foot mark on the long side at the top of the board. That will give you the rafter angle. Cut two rafters to those angles to go from the side of the addition to the peak. Leave space at the peak for a 2-inch ridge board and nail the rafters to studs in the house wall.
Cut and install three common rafters, one in the centre of the roof and one on each side sloping to the corners. Measure those spaces and use the table on the square to determine the length. It will show the length of rafter per foot of run, the distance covered. Nail those common rafters in place, the centre one fitting inside the peak of the rafters on the wall, the side rafters nailing to its corners.
Cut and install "jack" rafters along the sides, from the centre common rafter to overhang the wall edge. Use the table on the framing square to get the angle of cut. Jack rafters spaced 16 inches apart will decrease in length down the roof. Make a diagonal notch 1 inch deep and 4 inches long in the bottom of each rafter to sit firmly atop the wall cap. This is called a birds-mouth. Allow whatever overhang you want for the final rafter length.
Make jack rafters to parallel the centre common rafter from the side rafters to the end wall, also spaced 16 inches apart. Cut birds-mouth notches and the same overhang as on the side. When those are installed, cut short 2-by-4 joists to extend from the house wall to the end of the rafters. The side walls should already have this extension.
Deck the roof with oriented strand board (OSB) from the peaks to the outer walls. Nail metal flashing against the wall where the roof connects, one side up under the house siding, the other on the OSB. Nail metal flashing over the hip peaks and metal drip edge along all eaves. Lay down roofing paper with seams overlapped at least 2 inches and nail on shingles. Add a fascia or facing board along the ends of the rafters and a soffit cover to the joists, under the eaves, to finish the roof.
If joists on the addition are not covered, lay plywood or OSB on them for better footing.