7.3 Roof Geometry, Slope, Pitch & Slope Factors for Measuring Roofs

Basic Terms

Since span is twice the run, pitch = slope rise ÷ 24:

Measuring Slope on Site

Hold a level horizontally against the roof or rafter. Measure 12 inches along the level, then measure straight up to the roof. That measurement is the rise per foot. From drawings or known dimensions: slope = (rise ÷ run) × 12. A 14-foot run with a 7-foot rise is (7 ÷ 14) × 12 = 6:12.

The Common Slope Factor

The slope factor converts a horizontal distance to the distance along the slope:

$\text{Common factor} = \frac{\sqrt{\text{rise}^2 + 12^2}}{12}$

True area of a roof plane = horizontal (plan) area × common factor.

Rafter length (horizontal run including overhang) = run × common factor.

The Projection Rule

When every plane on a roof has the same slope, gable, hip, and cut-up roofs alike, the total roof area equals the total horizontal projected area, including overhangs, times the common factor. Measure the building footprint plus overhangs, multiply, and apply the factor. With different slopes, compute each area separately.

Hip and Valley Factors

On a regular roof, where planes of equal slope meet at 90-degree corners, hips and valleys run diagonally in plan. They travel about 16.97 inches horizontally for each 12 inches of common run:

$\text{Hip or valley factor} = \frac{\sqrt{\text{rise}^2 + 288}}{12}$

Hip or valley length = common run × hip or valley factor. You use these lengths to estimate ridge and hip caps, valley metal, and valley ice barrier.

Worked Problem 1: Gable Roof

A house is 28 feet by 46 feet, with 1-foot overhangs at both eaves and both rakes and a 6:12 slope.

1. Projected plan: (28 + 2) × (46 + 2) = 30 × 48 = 1,440 square feet.
2. True area: 1,440 × 1.118 = 1,609.9 square feet, or about 16.1 squares before waste.
3. Check with rafter length: run = 30 ÷ 2 = 15 feet, and 15 × 1.118 = 16.77 feet. Two planes: 2 × 48 × 16.77 = 1,609.9 square feet.
4. Rake length (each of 4 rakes) = 16.77 feet. Eave length = 48 feet × 2 = 96 feet.

Worked Problem 2: Hip Roof

A house is 32 feet by 56 feet, with 1.5-foot overhangs all around and an 8:12 slope.

1. Projected plan: 35 × 59 = 2,065 square feet.
2. True area: 2,065 × 1.202 = 2,482 square feet, or about 24.8 squares.
3. Common run = 35 ÷ 2 = 17.5 feet. Each hip = 17.5 × 1.563 = 27.4 feet. Four hips = about 109.4 feet.
4. Ridge length on a regular hip roof = length − width = 59 − 35 = 24 feet.
5. Hip plus ridge cap = 109.4 + 24 = about 133.4 linear feet.

Worked Problem 3: Converting a Measured Slope Length

From the eave, you measure 22 feet up a 10:12 slope to the ridge. The horizontal run is 22 ÷ 1.302 = 16.9 feet. The vertical rise is 16.9 × (10 ÷ 12) = 14.1 feet. Use this to check a drone report or plan against field measurements.

Measuring Existing Roofs

PSI lists three ways to measure: from blueprints and plans, from satellite imaging tools, and from the existing roof. For field measurement:

• Measure eaves, rakes, and each plane's slope length, and sketch the roof.
• Measure the slope of each plane. Dormers and porches often differ from the main roof.
• Note penetrations, valleys, hips, ridges, and wall lengths for flashing.
• Compare against the satellite or drone report. If they differ, trust careful field measurements.

Common Errors

• Forgetting the slope factor and ordering by plan area, which undercounts by 11.8% at 6:12 and 20.2% at 8:12.
• Using span instead of run for rafter or hip length.
• Leaving out overhangs.
• Applying the common factor to hip lengths. Use the hip or valley factor.