Why a hip rafter runs the diagonal — hip run is R√2, not common length × 1.414

A hip runs the diagonal of the run square, so its horizontal run is R√2. The correct length is √((R√2)² + rise²). Multiplying the finished common length by 1.414 is the common wrong shortcut — it overstates the hip about 5%.

A top-down plan of one hip bay drawn as a square whose side equals the common run R. A common rafter runs straight back one side of the square, a distance R. The hip rafter runs the diagonal of that square, so its horizontal run is R times the square root of two. The actual hip length is the square root of the diagonal run squared plus the rise squared, which equals R times the square root of two plus the pitch over twelve squared. The popular shortcut of multiplying the finished common rafter length by 1.414 is wrong — 1.414 applies to the horizontal run, not the sloped length — and it overstates the hip by roughly five percent. — Source: Traditional framing-square hip method (√2 diagonal run)

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A top-down plan of one hip bay drawn as a square whose side equals the common run R. A common rafter runs straight back one side of the square, a distance R. The hip rafter runs the diagonal of that square, so its horizontal run is R times the square root of two. The actual hip length is the square root of the diagonal run squared plus the rise squared, which equals R times the square root of two plus the pitch over twelve squared. The popular shortcut of multiplying the finished common rafter length by 1.414 is wrong — 1.414 applies to the horizontal run, not the sloped length — and it overstates the hip by roughly five percent.

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  <a href="https://remodelcalculators.com/construction-diagrams/rafter-hip-diagonal" target="_blank" rel="noopener">
    <img src="https://remodelcalculators.com/api/figure-embed/rafter-hip-diagonal" alt="A top-down plan of one hip bay drawn as a square whose side equals the common run R. A common rafter runs straight back one side of the square, a distance R. The hip rafter runs the diagonal of that square, so its horizontal run is R times the square root of two. The actual hip length is the square root of the diagonal run squared plus the rise squared, which equals R times the square root of two plus the pitch over twelve squared. The popular shortcut of multiplying the finished common rafter length by 1.414 is wrong — 1.414 applies to the horizontal run, not the sloped length — and it overstates the hip by roughly five percent." width="800" height="500" style="width:100%;height:auto;border:0" loading="lazy">
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    Diagram: <a href="https://remodelcalculators.com/construction-diagrams/rafter-hip-diagonal" target="_blank" rel="noopener">Why a hip rafter runs the diagonal — hip run is R√2, not common length × 1.414</a> — courtesy of <a href="https://remodelcalculators.com" target="_blank" rel="noopener">RemodelCalculators.com</a>
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