Heel Cut and Seat Cut Calculator for Rafters

This calculator determines a rafter's maximum allowable heel cut while ensuring the seat cut does not exceed IRC code limits. If the calculated seat cut surpasses the top plate width, the heel cut is automatically adjusted to comply with the code.

The heel cut is the vertical portion of the birdsmouth notch at the bottom of a rafter, where it rests on the top plate of a wall. It is part of the birdsmouth cut, which consists of:

• Heel Cut: The vertical cut determines how deep the rafter sits on the top plate.
• Seat Cut: The horizontal cut that rests on the top plate and distributes the load.

Birdsmouth Notch Limits

• To maintain structural integrity, the total birdsmouth notch depth (heel cut + seat cut) must not exceed 1/3 of the rafter depth.
  • Example: A 12-inch rafter can have a maximum notch depth of 4 inches.
• The seat cut cannot be larger than the width of the top plate to ensure proper bearing support.
  • Example: If the top plate is 3.5 inches wide, the seat cut must be 3.5 inches or less.

The calculator automatically meets these constraints by adjusting the heel cut if the seat cut exceeds the top plate width.

Calculator

1. enter the roof pitch, rise in 12 inches
2. the actual size of your rafter in inches
3. the width of the plate the rafter rests on
4. fraction precision, preset at 16

Results

1. the longest possible vertical heel cut while staying within code limits
2. the seat cut based on the maximum heel cut
3. the adjusted seat cut if it exceeds the top plate width
4. the recalculated heel cut when the seat cut is reduced

Heel Cut Length Calculator - Formulas and Example Calculation

Inputs:

• Roof Pitch = 8
• Rafter Depth = \(11 \frac{7}{8}\) in
• Top Plate Width = \(5 \frac{1}{2}\) in
• Fraction Precision = 16

Formulas Used

Slope Calculation

\[ \text{slope} = \frac{\sqrt{\text{pitch}^2 + 12^2}}{12} \]

Maximum Heel Cut Length

\[ \text{heel cut}_{\text{max}} = \frac{\text{rafter depth}}{3} \]

Initial Seat Cut Calculation (Before Adjustment)

\[ \text{seat cut}_{\text{initial}} = \frac{\text{heel cut}_{\text{max}}}{\text{slope}} \]

Heel Cut Calculation

\[ \text{heel cut} = \sqrt{(\text{seat cut} \times \text{slope})^2 - \text{seat cut}^2} \]

Notch Size Calculation

\[ \text{notch size} = \sqrt{\text{seat cut}^2 + \text{heel cut}^2} \]

Adjusted Seat Cut (If Exceeding Top Plate Width)

\[ \text{seat cut}_{\text{final}} = \min(\text{seat cut}_{\text{initial}}, \text{top plate width}) \]

Adjusted Heel Cut

\[ \text{heel cut}_{\text{adjusted}} = \sqrt{(\text{seat cut}_{\text{final}} \times \text{slope})^2 - \text{seat cut}_{\text{final}}^2} \]

Step-by-Step Example Calculation

Step 1: Compute Slope Factor

\[ \text{slope} = \frac{\sqrt{8^2 + 12^2}}{12} = \frac{\sqrt{64 + 144}}{12} = \frac{\sqrt{208}}{12} \approx 1.198 \]

Step 2: Compute Maximum Heel Cut

\[ \text{heel cut}_{\text{max}} = \frac{11.875}{3} \approx 3.96 \text{ in} \]

Step 3: Compute Initial Seat Cut

\[ \text{seat cut}_{\text{initial}} = \frac{3.96}{1.198} \approx 3.31 \text{ in} \]

Step 4: Compute Heel Cut

\[ \text{heel cut} = \sqrt{(3.31 \times 1.198)^2 - 3.31^2} \] \[ = \sqrt{(3.97)^2 - 3.31^2} \] \[ = \sqrt{15.76 - 10.96} = \sqrt{4.80} \approx 2.19 \text{ in} \]

Step 5: Compute Notch Size

\[ \text{notch size} = \sqrt{3.31^2 + 2.19^2} \] \[ = \sqrt{10.96 + 4.80} = \sqrt{15.76} \approx 3.97 \text{ in} \]

Step 6: Adjust Seat Cut (If Exceeding Top Plate Width)

Since: \[ \text{seat cut}_{\text{initial}} = 3.31 \text{ in} < \text{top plate width} = 5.5 \text{ in} \] No adjustment needed: \[ \text{seat cut}_{\text{final}} = 3.31 \text{ in} \]

Final Results

\[ \begin{aligned} \text{Heel Cut Length} &= 2 \frac{3}{16} \text{ in} \\ \text{Initial Seat Cut Width (Before Adjustment)} &= 3 \frac{5}{16} \text{ in} \\ \text{Final Seat Cut Width} &= 3 \frac{5}{16} \text{ in} \\ \text{Adjusted Heel Cut (if seat cut exceeds top plate width)} &= 2 \frac{3}{16} \text{ in} \end{aligned} \]

Step-by-Step Example Calculation (With Adjustment)

Inputs:

• Roof Pitch = 5
• Rafter Depth = \(11 \frac{7}{8}\) in
• Top Plate Width = \(3 \frac{1}{2}\) in
• Fraction Precision = 16

Step 1: Compute Slope Factor

\[ \text{slope} = \frac{\sqrt{5^2 + 12^2}}{12} = \frac{\sqrt{25 + 144}}{12} = \frac{\sqrt{169}}{12} = \frac{13}{12} \approx 1.083 \]

Step 2: Compute Maximum Heel Cut

\[ \text{heel cut}_{\text{max}} = \frac{11.875}{3} \approx 3.96 \text{ in} \]

Step 3: Compute Initial Seat Cut

\[ \text{seat cut}_{\text{initial}} = \frac{3.96}{1.083} \approx 3.625 \text{ in} = 3 \frac{5}{8} \text{ in} \]

Step 4: Compute Heel Cut

\[ \text{heel cut} = \sqrt{(3.625 \times 1.083)^2 - 3.625^2} \] \[ = \sqrt{(3.92)^2 - 3.625^2} \] \[ = \sqrt{15.37 - 13.14} = \sqrt{2.23} \approx 1.50 \text{ in} = 1 \frac{1}{2} \text{ in} \]

Step 5: Compute Notch Size

\[ \text{notch size} = \sqrt{3.625^2 + 1.50^2} \] \[ = \sqrt{13.14 + 2.25} = \sqrt{15.39} \approx 3.92 \text{ in} \]

Step 6: Adjust Seat Cut (If Exceeding Top Plate Width)

Since: \[ \text{seat cut}_{\text{initial}} = 3 \frac{5}{8} \text{ in} > \text{top plate width} = 3 \frac{1}{2} \text{ in} \] Adjustment required: \[ \text{seat cut}_{\text{final}} = 3 \frac{1}{2} \text{ in} \]

Step 7: Compute Adjusted Heel Cut

\[ \text{heel cut}_{\text{adjusted}} = \sqrt{(3.5 \times 1.083)^2 - 3.5^2} \] \[ = \sqrt{(3.79)^2 - 3.5^2} \] \[ = \sqrt{14.37 - 12.25} = \sqrt{2.12} \approx 1.44 \text{ in} = 1 \frac{7}{16} \text{ in} \]

Final Results (With Adjustment)

\[ \begin{aligned} \text{Heel Cut Length (Before Adjustment)} &= 1 \frac{1}{2} \text{ in} \\ \text{Initial Seat Cut Width (Before Adjustment)} &= 3 \frac{5}{8} \text{ in} \\ \text{Final Seat Cut Width (After Adjustment)} &= 3 \frac{1}{2} \text{ in} \\ \text{Adjusted Heel Cut (if seat cut exceeds top plate width)} &= 1 \frac{7}{16} \text{ in} \end{aligned} \]