# Dachneigung berechnen: Winkel, Gefälle und Steigung

> Aus Höhe und waagerechter Strecke Winkel, Steigung in Prozent und Dachneigung (x zu 12) sowie schräge Länge und Sparrenfaktor berechnen.

Interaktive Version: https://www.calcopenly.com/de/geometry/slope-roof-pitch-calculator
Thema: Geometrierechner

Every way of stating a slope comes from one ratio, m = rise ÷ run. The angle from horizontal is arctan m, the percent grade is 100m, the roof pitch is 12m (units of rise per 12 of run), and the rafter factor √(1 + m²) turns a horizontal run into the sloped length. Enter a rise and run, or any one of the other forms, and the rest follow.

The default rise of 6 over a run of 12 is a 6/12 roof: 26.57°, a 50% grade and a rafter factor of 1.1180, so 12 cm of run needs 13.4164 cm of rafter. Ramps and roads use the same numbers: the steepest ramp the 2010 ADA Standards allow, 1:12, is an 8.33% grade, or 4.76°.

Percent grade is not an angle: a 100% grade is 45°, and a vertical face has no finite grade or pitch, so angles must lie between −90° and 90°. A negative rise describes a downhill slope.

## Eingaben

- **Bekannte Größe** (Optionen: Rise and run, Winkel, Percent grade, Roof pitch)
- **Rise**: Negative for a downhill slope
- **Run (horizontal distance)**
- **Angle from horizontal**: Between −90° and 90°
- **Note**: Rise per 100 of run
- **Pitch**: Rise for every 12 of run, as in a 6/12 roof
- **Ergebniseinheit** (Optionen: Millimeter (mm), Zentimeter (cm), Meter (m), Kilometer (km), Zoll (in), Fuß (ft), Yard (yd), Meilen (mi))

## Ergebnisse

- Winkel (°) — Hauptergebnis
- Note
- Roof pitch (in 12)
- Ratio (rise : run)
- Rise per unit of run
- Rafter length factor
- Slope length

## Formel

$$
\begin{gathered} m = \frac{\text{rise}}{\text{run}},\quad \theta = \arctan m,\quad \text{grade} = 100m\,\% \\[6pt] \text{pitch} = 12m \text{ in } 12,\quad \text{factor} = \sqrt{1 + m^2} \end{gathered}
$$

## Durchgerechnete Beispiele

### Rise 6, run 12 (a 6/12 roof)

- Bekannte Größe: Rise and run
- Rise: 6 cm
- Run (horizontal distance): 12 cm
- Ergebniseinheit: Zentimeter (cm)
- **Winkel: 26.565051 °**
- **Note: 50%**
- **Roof pitch: 6 in 12**
- **Ratio (rise : run): 1 : 2**
- **Rafter length factor: 1.118034**
- **Slope length: 13.416408 cm**
- Prüfquelle: Python 3.8 math: degrees(atan(0.5)), sqrt(1.25), sqrt(6**2 + 12**2)

### 45°

- Bekannte Größe: Winkel
- Angle from horizontal: 45 °
- **Note: 100%**
- **Roof pitch: 12 in 12**
- **Ratio (rise : run): 1 : 1**
- **Rafter length factor: 1.414214**
- Prüfquelle: tan 45° = 1; Python 3.8 math.sqrt(2)

### 8 % road grade

- Bekannte Größe: Percent grade
- Note: 8%
- **Winkel: 4.573921 °**
- **Roof pitch: 0.96 in 12**
- **Ratio (rise : run): 1 : 12.5**
- Prüfquelle: Python 3.8 math: degrees(atan(0.08)); 12 × 0.08; 1/0.08

### 4/12 roof pitch

- Bekannte Größe: Roof pitch
- Pitch: 4 in 12
- **Winkel: 18.434949 °**
- **Note: 33.333333%**
- **Rafter length factor: 1.054093**
- Prüfquelle: Python 3.8 math: degrees(atan(4/12)), sqrt(1 + 1/9)

### ADA ramp 1:12

- Bekannte Größe: Rise and run
- Rise: 1 cm
- Run (horizontal distance): 12 cm
- Ergebniseinheit: Zentimeter (cm)
- **Note: 8.333333%**
- **Winkel: 4.763642 °**
- **Ratio (rise : run): 1 : 12**
- Prüfquelle: ADA §405.2 maximum ramp slope 1:12; Python 3.8 math: degrees(atan(1/12))

### Flat ground (edge case)

- Bekannte Größe: Rise and run
- Rise: 0 cm
- Run (horizontal distance): 5 cm
- Ergebniseinheit: Zentimeter (cm)
- **Winkel: 0 °**
- **Note: 0%**
- **Roof pitch: 0 in 12**
- **Rafter length factor: 1**
- **Slope length: 5 cm**
- Prüfquelle: Zero rise: tan θ = 0

## Fragen

### How do you convert percent grade to degrees?

Take the arctangent of the grade divided by 100: θ = arctan(grade/100). An 8% road grade is arctan 0.08 = 4.57°, and a 100% grade is 45°, not 90°. The reverse is grade = 100 × tan θ, so a 30° slope is a 57.7% grade.

### What angle is a 6/12 roof pitch?

26.57°. A pitch of x/12 means x units of rise for every 12 of run, so the angle is arctan(x/12): 4/12 is 18.43°, 6/12 is 26.57°, 8/12 is 33.69° and 12/12 is 45°. The rafter factor for 6/12 is √1.25 = 1.118, so 12 ft of horizontal run needs 13.42 ft of rafter before any overhang.

### What is the maximum slope for a wheelchair ramp?

1:12 under the 2010 ADA Standards for Accessible Design (§405.2): one unit of rise for every 12 of run, an 8.33% grade or 4.76°. Each ramp run may rise at most 30 in (§405.6), so a 30 in rise needs a run of at least 30 ft. Other countries set their own limits in their building codes.

### How do you calculate rise over run?

Divide the vertical rise by the horizontal run, both in the same unit: m = rise/run. A roof that rises 6 in over 12 in of run has m = 0.5. Measure the run horizontally, not along the surface: the sloped length here is 13.42 in, and dividing by it gives 0.447, the sine of the angle rather than the slope.

### Wie genau arbeitet „Dachneigung berechnen: Winkel, Gefälle und Steigung“?

Die Genauigkeit hängt von Ihren Eingaben und den Annahmen der Methode ab. Die Dezimalrechnung nutzt 50 signifikante Stellen, doch Schätzungen, numerische Verfahren und Quelldaten können ungenauer sein. Die angezeigte Rundung beseitigt diese Grenzen nicht. Anhand unabhängiger Quellen geprüfte Rechenbeispiele: 8. Beispielsweise wird „Rise 6, run 12 (a 6/12 roof)“ anhand von Python 3.8 math: degrees(atan(0.5)), sqrt(1.25), sqrt(6**2 + 12**2) geprüft.

### Woher stammt die Methode?

OpenStax College Algebra 2e, §4.1 — slope as rise over run; US Access Board, ADA Standards §405.2 — ramp slope 1:12 (8.33 %).

## Quellen

- [OpenStax College Algebra 2e, §4.1 — slope as rise over run](https://openstax.org/books/college-algebra-2e/pages/4-1-linear-functions)
- [US Access Board, ADA Standards §405.2 — ramp slope 1:12 (8.33 %)](https://www.access-board.gov/ada/#ada-405)