During my 30 years as a carpenter, I have framed several hundred complicated roofs, and have read nearly every book on the subject. A particularly good one is The Steel Square, by H.H. Siegele, which uses drawing techniques rather than advanced math to illustrate complex rafter cuts.

I began using these drawing techniques because I wanted to better understand the geometry of roof planes. Not only did these methods help me grasp the way complex roofs fit together, but they also increased my cutting speed — and have been excellent teaching tools to boot.

In this article, I'll show how I use this technique to develop the basic cuts for an irregular hip. The method explained here is accurate for concealed 2-by roof framing. Where accuracy is more critical — as in the timber-frame homes we build, where a half-degree error on an 8/12 timber valley really shows — I also use basic trigonometry, but that's another story.

When teaching framing to novice carpenters, the author uses a simple cardboard model to help them "think in triangles."

### Thinking in Triangles

This method helps you think of a roof in terms of its component triangles and how they relate to each other. All the triangles that make up the roof are drawn on a flat surface. Think of the horizontal leg of each triangle — the run — as a hinge. Once the drawing is complete, the triangles can be mentally rotated on their hinges to form a three-dimensional model of the roof. When I teach this method, I use a cardboard model as a visual aid to show how each part of the roof relates to the others. Mastering the technique takes time — I'm still learning — but the basics can be grasped quite easily.

### Plywood Worksheet

I usually make my drawings on a piece of plywood, using the 12ths scale on the framing square. This makes sense on the job site, and lets me draw the cut angles full-scale. The only special tools I use are a large compass (Lee Valley Tools sells one; 800/267-8735, www.leevalley.com) and an oversized bevel square. The aluminum bevel square in the photos came from Germany, but any large one will work. (Quint Measuring Systems' contractor-grade True Angle Tool is available online in 24-inch and 36-inch lengths; www.quintmeasuring.com.)

The blueprints I'm typically given include only the most basic information about a roof: a plan view, an elevation, and the basic pitches. But this is enough information to develop the drawings, if you follow the correct steps. The easiest method is to break the drawing sequence down into discrete parts, each of which builds on the previous one:

• plan angle for the hip rafter

• common and hip lengths and cut angles

• hip backing angle

• roof surface

### Draw the Common and Hip Runs

The example roof has a 5/12 pitch on the hip end and a 10/12 pitch on the building's long dimension, and the eaves meet at a 90-degree angle. Note that I'm ignoring the overhang and working from the plate line.

• The full-scale drawing for an irregular hip roof begins with a basic rectangle — a plan view that represents a corner of the roof. The sides are the two common runs; the diagonal is the hip rafter.

To get started, you first have to accurately draw the angle at which the hip rafter intersects the eaves at the corner. I start with an imaginary point on the hip rafter where the rise is 12 inches above the outside of the plate, then determine the run of the common rafters from that point to the plate line. You can do this quickly with a calculator, but I just use a simple table I've printed out.

• To start the full-scale layout drawing, the author finds the two common runs at 12 inches of rise; that is, he converts the 5-inch-over-12-inch roof pitch to 12 inches over 28 13/16 inches.

Rather than dothe math each time, he works from a chart.

A 5/12 common rafter has a run of 28 13/16 inches at 12 inches of rise, while the 10/12 rafter runs 14 3/8 inches. I use these numbers to draw a rectangle, which is the basis for all the drawings to follow. A diagonal across the middle (line BD) represents the hip in plan. If you've drawn accurately, you've already got the side cut angles for the jacks, which you can transfer to the stock with a bevel square.