Birdsmouth Seat Cut Calculator

This calculator determines the seat cut, notch size, and remaining rafter depth for a birdsmouth joint in roof framing. It calculates the correct horizontal seat cut and notch size (hypotenuse) based on the roof pitch, heel cut, wall width, and rafter depth.

A birdsmouth cut is a notch in a rafter that allows it to sit securely on the wall plate in roof framing. It

consists of two cuts:

• Seat Cut (Horizontal): Rests on the wall plate to provide a stable bearing surface.
• Heel Cut (Vertical): Fits against the wall to maintain alignment.

The seat cut must not exceed the wall plate width, and the notch size must not exceed 1/3 of the rafter depth to ensure structural integrity.

Calculator

1. enter the vertical rise of the roof per 12 inches of horizontal run
2. the length of the heel cut in inches (e.g. 3 1/2)
3. the width of the top wall plate in inches
4. the size of the rafter in inches (e.g. 9 3/8)

Results

1. the length of the seat cut in inches
2. the vertical depth of the cut into the rafter (notch depth) in inches
3. the remaining depth of the rafter measured from the top of the rafter to the deepest point of the birdsmouth notch into the rafter in inches

Birdsmouth Seat Cut Calculator - Formulas

The birdsmouth cut is calculated using trigonometric formulas based on the roof pitch and rafter dimensions.

1. Roof Pitch Angle Calculation

The pitch angle \( \theta \) is calculated using the roof rise per 12 inches of run: \[ \theta = \tan^{-1} \left( \frac{\text{Rise}}{12} \right) \]

2. Seat Cut Calculation

The seat cut length is calculated as: \[ \text{Seat Cut} = \frac{\text{Heel Cut}}{\tan(\theta)} \]

3. Notch Size Calculation

The notch size is the horizontal projection of the heel cut: \[ \text{Notch Size} = \text{Heel Cut} \times \cos(\theta) \]

4. Remaining Depth Calculation

The remaining rafter depth after the notch is cut: \[ \text{Remaining Depth} = \text{Rafter Depth} - \text{Notch Size} \]

5. Height Above Plate Calculation

The height above the wall plate is: \[ \text{Height Above Plate} = \frac{\text{Rafter Depth}}{\cos(\theta)} - \text{Heel Cut} \]

Example Calculation with Mixed Fractions

Given the following values:

• Roof Pitch: \(6:12\) (Rise = 6 inches per 12 inches run)
• Heel Cut: \(1\) inch
• Wall Width: \(5 \frac{1}{2}\) inches
• Rafter Depth: \(11 \frac{7}{8}\) inches

Calculations:

• Pitch Angle: \[ \theta = \tan^{-1} \left( \frac{6}{12} \right) = \tan^{-1}(0.5) \approx 26.57^\circ \]
• Seat Cut: \[ \text{Seat Cut} = \frac{1}{\tan(26.57^\circ)} = \frac{1}{0.5} = 2 \text{ inches} \]
• Notch Size: \[ \text{Notch Size} = 1 \times \cos(26.57^\circ) = 1 \times 0.894 = \frac{7}{8} \text{ inches} \]
• Remaining Depth: \[ \text{Remaining Depth} = 11.875 - 0.875 = 11 \text{ inches} \]
• Height Above Plate: \[ \text{Height Above Plate} = \frac{11.875}{\cos(26.57^\circ)} - 1 = \frac{11.875}{0.894} - 1 \approx 12 \frac{1}{4} \text{ inches} \]

Point-Form Summary

• Seat Cut: 2 inches
• Notch Size: 7/8 inches
• Remaining Depth: 11 inches
• Height Above Plate: 12 1/4 inches