Heel Cut Length Calculator - Metric (Rafter Depth and Top Plate Width in mm)

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

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

Heel Cut Length Calculator - Corrected Math Formulas

Example 1: Using Roof Pitch in Rise over 12

Inputs:

• Roof Pitch = 8 in 12
• Rafter Depth = 235 mm
• Top Plate Width = 140 mm

Step 1: Convert Roof Pitch to Degrees

\[ \theta = \tan^{-1}\left(\frac{\text{rise}}{12}\right) \] \[ \theta = \tan^{-1}\left(\frac{8}{12}\right) = \tan^{-1}(0.6667) \approx 33.69^\circ \]

Step 2: Calculate Roof Slope

\[ \text{slope} = \frac{\sqrt{rise^2 + run^2}}{run} \] \[ \text{slope} = \frac{\sqrt{8^2 + 12^2}}{12} = \frac{\sqrt{64 + 144}}{12} = \frac{\sqrt{208}}{12} = 1.20185 \]

Step 3: Calculate Maximum Notch Depth

\[ \text{maxNotch} = \frac{\text{rafter depth}}{3} = \frac{235}{3} = 78.3333 \ \text{mm} \]

Step 4: Initial Seat Cut Width

\[ \text{initialSeatCut} = \frac{\text{maxNotch}}{\text{slope}} = \frac{78.3333}{1.20185} = 65.18 \ \text{mm} \]

Step 5: Heel Cut Length

\[ \text{heelCut} = \sqrt{(initialSeatCut \cdot slope)^2 - initialSeatCut^2} \] \[ = \sqrt{(65.18 \cdot 1.20185)^2 - 65.18^2} = \sqrt{(78.3333)^2 - (65.18)^2} = \sqrt{6135.14 - 4259.77} = \sqrt{1875.37} = 43.45 \ \text{mm} \]

Step 6: Final Seat Cut and Adjusted Heel Cut

If: \[ \text{topPlateWidth} = 140 \ \text{mm} \] Since: \[ 65.18 \leq 140 \] The seat cut fits, so no adjustment is needed. Final outputs:

• Heel Cut Length = \(43.45 \ \text{mm}\)
• Initial Seat Cut Width = \(65.18 \ \text{mm}\)
• Final Seat Cut Width = \(65.18 \ \text{mm}\)
• Adjusted Heel Cut Length = \(43.45 \ \text{mm}\)

---

Example 2: Using Roof Pitch in Degrees

Inputs:

• Roof Pitch = 33.69°
• Rafter Depth = 235 mm
• Top Plate Width = 140 mm

Step 1: Convert Degrees to Rise over 12

\[ \text{rise} = 12 \cdot \tan(\theta) \] \[ \text{rise} = 12 \cdot \tan(33.69°) = 12 \cdot 0.6667 = 8.0004 \approx 8 \ \text{inches} \] This confirms the pitch converts correctly back to rise over 12.

Step 2: Repeat Steps 2 to 6 from Example 1

Since the slope and dimensions are the same, the rest of the process (seat cut, heel cut) produces the exact same results:

• Heel Cut Length = \(43.45 \ \text{mm}\)
• Initial Seat Cut Width = \(65.18 \ \text{mm}\)
• Final Seat Cut Width = \(65.18 \ \text{mm}\)
• Adjusted Heel Cut Length = \(43.45 \ \text{mm}\)

---

Summary of Key Formulas

• Convert rise over 12 to slope: \[ \text{slope} = \frac{\sqrt{rise^2 + run^2}}{run} \]
• Convert rise over 12 to degrees: \[ \theta = \tan^{-1}\left(\frac{\text{rise}}{12}\right) \]
• Convert degrees to rise over 12: \[ \text{rise} = 12 \cdot \tan(\theta) \]
• Initial seat cut width: \[ \frac{\text{maxNotch}}{\text{slope}} \]
• Heel cut length: \[ \sqrt{(initialSeatCut \cdot slope)^2 - initialSeatCut^2} \]
• Maximum notch: \[ \frac{\text{rafter depth}}{3} \]