Roof geometry
Hip or valley rafter length and pitch (equal pitches, square corner)
- where
- plan length of the hip (mm)
- run of the common rafter (mm)
- rise (mm)
- roof pitch (°)
- hip or valley rafter length (mm)
- pitch of the hip or valley (°)
- hip length factor per unit of common run
Explain this equation
Gives the length and pitch of a hip or valley rafter where two slopes of the same pitch meet over a square corner. Use it to cut hips and valleys and to order the timber or the hip tiles.
On plan a hip runs diagonally across a square whose sides are the common run G, so its plan length H is G times √2. It climbs the same rise R as the common rafters, but over that longer plan distance. L_h is the true length along the hip, and p_h is its shallower pitch. k is a handy multiplier: hip length per unit of common run.
The hip sits in a right-angled triangle with base H and height R, so Pythagoras gives its length and arctan gives its angle.
Sense check: a 3 m run at 30° gives a 4.24 m plan length, a 1.73 m rise and a hip of about 4.58 m at about 22°. The hip is always flatter than the main slopes.
Watch out: this only holds for equal pitches meeting at 90° on plan. Unequal pitches or splayed corners need a different set-out.
True length and pitch of a hip or valley rafter where two slopes of the same pitch meet over a square (90°) corner. In plan the hip runs at 45°.
Enter your values to see the result.
Good to know
The hip rises the same height as the common rafters but over a longer plan distance (the diagonal, √2 times the run), so it is longer and flatter. Hip and valley tiles and dry hip systems have their own minimum pitches; a 30° roof has a hip of only about 22°. Use the same run definition as the common rafter (to the centre line) and adjust for the ridge and hip thicknesses when cutting.
Common mistake
Assuming the hip has the same pitch as the roof. It is always flatter, which matters for bonnet hip tiles and valley trough minimum pitches.
Limits
Equal pitches and a 90° plan corner only. Unequal pitches or splayed corners need the full geometry. Overhang: include it in the run to get the length to the eaves corner.
Sources
Results are calculations from the published methods and figures shown, and depend on the data you enter. Follow the roof covering manufacturer’s fixing specification, BS 5534, BS 8000-6, BS 6229, the Lead Sheet Training Academy guidance and Building Regulations. Wind uplift and structural design need a competent designer.