Direct answer
Slope is rise divided by horizontal run; percentage grade multiplies that ratio by 100, while the angle is arctan(rise ÷ run).
What this calculation tells you
Rise-and-run calculations describe the inclination and sloped length between two points. The same geometry can be expressed as a ratio, grade, angle, fall per unit length, or pitch, but the notation must be stated because different trades prefer different conventions.
The calculation supports layout and communication; it does not establish whether the inclination is suitable or compliant for a particular structure, surface, route, or drainage system.
Where it is used
Roofing and carpentry
Relate roof rise, horizontal run, rafter length, cut geometry, and pitch notation during layout and estimating.
Drainage and plumbing
Communicate pipe or surface fall between known elevations while checking the applicable design criteria separately.
Roads, ramps, and site work
Describe grade, elevation change, and sloped distance for preliminary layout, surveying, or quantity work.
Fabrication and installation
Lay out braces, rails, frames, conveyors, or equipment transitions from measured offsets and angles.
Common situations
- Converting a measured elevation difference and horizontal distance into percentage grade.
- Finding a diagonal length for a brace, ramp, rafter, or sloped component.
- Translating between an angle and a trade-specific ratio without changing the geometry.
- Writing field instructions that make the direction of fall and datum unambiguous.
Four related outputs
A rise of 3 over a horizontal run of 12 has a ratio of 3:12, which simplifies to 1:4. The grade is 25%, and the angle is arctan(0.25), about 14.04°.
The sloped length is the hypotenuse: √(rise² + run²). For the same example it is √153, about 12.37 in the same length unit.
Do not mix the denominator
Percentage grade uses horizontal run as its denominator. Dividing rise by the diagonal produces a sine ratio instead and answers a different question.
Measurements must use compatible units before division. A rise in millimetres and run in metres must be converted to a common unit.
Professional limits
The geometry does not decide whether a slope complies with drainage, access, roof, roadway, or safety requirements. Verify the applicable design documents, site conditions, and local requirements.
- Measure rise vertically and run horizontally.
- Name the convention beside the result.
- Use adequate significant figures for layout, then follow field tolerances.
Communicating slope without ambiguity
Different teams may describe the same inclination as a ratio, percentage, angle, fall per unit length, or roof-pitch notation. A number without its convention can therefore be interpreted several ways. Drawings and field notes should state both the value and the convention, such as “2% fall” or “1:50 downward.”
Direction matters too. A positive mathematical slope may be described on site as fall in the direction of flow. Name the high point, low point, datum, and direction so that a correct magnitude is not laid out backward.
- Include units and direction on every field note.
- Do not assume a ratio follows the same ordering convention in every trade.
- Use two-point elevations when the datum matters.
Practical questions
Frequently asked questions
Is run the same as sloped length?
No. Run is the horizontal projection. Sloped length is the diagonal between the endpoints and is longer whenever the rise is nonzero.
What angle is a 100% grade?
A 100% grade means rise equals horizontal run, which corresponds to a 45° angle from level.
Can a calculated slope confirm compliance?
No. It provides geometry only. Applicable drawings, codes, accessibility criteria, drainage design, product requirements, and site tolerances must be checked separately.