To work out the true length of a hip rafter, you need to understand its relationship to a common rafter and the roof's geometry. The most straightforward method involves using the hip rafter's "unit length" in conjunction with the common rafter's run.
A hip rafter is a structural timber that extends from the corner of a building to the ridge, forming the hip (outside) corner of a hip roof. It supports hip jack rafters and defines the sloped edges of the roof plane. Hip rafters are distinct from common rafters, which run perpendicular to the wall plate up to the ridge board. Hip rafters are inherently longer than common rafters with the same rise because they cover a greater horizontal distance diagonally across the corner of the building.
The Core Formula for Hip Rafter Length
As per common roofing calculations, to determine the true length of a hip rafter, you can use a simple multiplication:
Multiply the unit length of the hip rafter by the number of feet in the total run of a common rafter.
Let's break down the components of this formula:
-
Unit Length of a Hip Rafter: This is the actual length of the hip rafter for every 1 foot (12 inches) of common rafter run. It accounts for both the roof's rise (vertical height) and the hip rafter's diagonal horizontal projection.
- To calculate this unit length, you typically consider a horizontal run of 17 inches for the hip rafter (which is the diagonal of a 12-inch by 12-inch square, approximately 16.97 inches, often rounded to 17 inches for practical purposes) and the actual rise per foot of the common rafter.
- The formula to find the unit length of a hip rafter is:
$ \text{Unit Length}_{\text{hip}} = \sqrt{(\text{Hip Unit Run})^2 + (\text{Common Rafter Rise per Foot})^2} $
Where Hip Unit Run $\approx 17$ inches.
-
Total Run of a Common Rafter: This is the horizontal distance measured from the outside face of the wall plate (or building line) to the plumb line of the ridge board. This measurement is crucial as it forms the base of the triangle used in calculating rafter lengths.
Step-by-Step Calculation
Here’s how to apply the formula in practice:
- Determine the Roof Pitch: Identify the roof's rise per foot of run (e.g., 6/12 pitch means 6 inches of rise for every 12 inches of horizontal run).
- Calculate the Unit Length of the Hip Rafter:
- Use the common rafter's rise per foot.
- Use the hip rafter's unit run (approximately 16.97 or 17 inches).
- Apply the Pythagorean theorem: $\sqrt{(\text{16.97 inches})^2 + (\text{Rise per foot})^2}$.
- Measure the Total Run of the Common Rafter: Find the horizontal distance from the building line to the ridge in feet.
- Apply the Formula: Multiply the calculated unit length (from Step 2, ensuring consistent units, often converted to feet for the final calculation if the unit length is in inches) by the total run of the common rafter (in feet).
Formula Summary:
$ \text{Hip Rafter Length (in feet)} = \text{Hip Rafter Unit Length (in inches/foot of common run)} \times \text{Total Common Rafter Run (in feet)} / 12 \text{ (to convert inches to feet)} $
Or, if the Hip Rafter Unit Length is already calculated per foot of common run:
$ \text{Hip Rafter Length} = \text{Hip Unit Length (in inches per foot of run)} \times \text{Total Common Rafter Run (in feet)} $
Example Calculation
Let's say you have a roof with an 8/12 pitch (8 inches of rise per 12 inches of run) and the total run of the common rafter is 10 feet.
- Roof Pitch: 8/12 (Rise = 8 inches per foot).
- Calculate Hip Rafter Unit Length:
- Hip Unit Run $\approx 16.97$ inches.
- Rise = 8 inches.
- Unit Length = $\sqrt{16.97^2 + 8^2} = \sqrt{287.98 + 64} = \sqrt{351.98} \approx 18.76$ inches.
- This means for every 1 foot of common rafter run, the hip rafter adds 18.76 inches to its length.
- Total Common Rafter Run: 10 feet.
- Apply the Formula:
- Hip Rafter Length = $18.76 \text{ inches/foot} \times 10 \text{ feet} = 187.6 \text{ inches}$.
- To convert to feet: $187.6 \text{ inches} / 12 \text{ inches/foot} \approx 15.63 \text{ feet}$.
So, the true length of the hip rafter would be approximately 15.63 feet.
Hip Rafter Unit Lengths for Various Pitches
The following table illustrates typical unit lengths for common and hip rafters based on different roof pitches:
| Roof Pitch (Rise/Run) | Common Rafter Unit Length (per 12" run) | Hip Rafter Unit Length (per 12" common rafter run) |
|---|---|---|
| 4/12 | $\sqrt{12^2 + 4^2} \approx 12.65"$ | $\sqrt{16.97^2 + 4^2} \approx 17.43"$ |
| 6/12 | $\sqrt{12^2 + 6^2} \approx 13.42"$ | $\sqrt{16.97^2 + 6^2} \approx 17.98"$ |
| 8/12 | $\sqrt{12^2 + 8^2} \approx 14.42"$ | $\sqrt{16.97^2 + 8^2} \approx 18.76"$ |
| 10/12 | $\sqrt{12^2 + 10^2} \approx 15.62"$ | $\sqrt{16.97^2 + 10^2} \approx 19.71"$ |
| 12/12 | $\sqrt{12^2 + 12^2} \approx 16.97"$ | $\sqrt{16.97^2 + 12^2} \approx 20.88"$ |
Key Considerations for Accuracy
- Roof Pitch Accuracy: Precise measurement of the roof's rise and run is critical. Even slight errors can lead to significant discrepancies in rafter lengths.
- Plumb and Level Lines: All measurements for run and rise must be taken from true plumb and level lines.
- Ridge Board/Hip Purlin Deduction: The calculated length is the theoretical "line length." In practice, you must deduct half the thickness of the ridge board (where the hip rafter meets the ridge) and any hip purlins if they are used, to get the actual cutting length for the rafter.
- Compound Angles: As mentioned, hip rafter cuts (especially for the seat cut and bird's mouth) are typically made at compound angles, requiring precise layout and cutting skills. These angles are calculated based on the roof pitch.
- Building Corners: Ensure the building corners are truly square (90 degrees) for the 17-inch hip unit run approximation to be accurate.
Tools and Resources
- Framing Square: An indispensable tool for roof framing, often featuring rafter tables or scales to assist in calculations.
- Construction Calculator: Many specialized calculators have built-in rafter functions that simplify these calculations.
- Rafter Tables: Pre-calculated tables for various pitches can be found in carpentry handbooks or online, providing unit lengths and angles.
