App to calculate a slope: roof, terrain and staircase, from the formula in PDF format

An application for calculating a slope meets one of three needs: obtaining a slope in percentage or degrees from already known coordinates, measuring the actual inclination of a surface with a sensor, or measuring a slope on a plan. MeasureAll meets the third need: the application uses your PDF plans on iPhone, iPad, or Android and, when the plan is reliable, the measurement can be taken at the office, without having to climb onto a roof for the only point measurement. In all cases, the calculation is based on the same formula.

This page details it with its conversions, then applies it to three concrete cases: the roof and the length of its ramp, the terrain, and the staircase. It finally helps to choose the right tool and shows how to plot these values on a PDF.

Calculating a slope: formula and conversions

Slope compares a vertical ascent, the gradient, to the horizontal distance traveled. Expressed as a percentage, it is equal to:

Slope (%) = 100 × elevation ÷ horizontal distance

The two lengths must be in the same unit, and the distance must be horizontal, not measured along the slope. A climb of 0.20 m on 2.00 m of horizontal ground gives 100 × 0.20 ÷ 2.00 = 10%. This is the same slope as 0.10 m on 1.00 m: doubling the two lengths does not change their ratio. This Definition of the slope, recalled by Éduscol, applies to a roof as well as to a plot of land.

To convert to degrees, take the arctangent of the ratio: angle = arctan(slope ÷ 100). In reverse, slope = 100 × tan(angle). A slope of 100% therefore does not correspond to 90°, but to 45°: the elevation then equals the horizontal distance.

Slope	Lecture	Angle
1 %	1 cm per meter horizontally	0,57°
5 %	5 cm per meter horizontally	2,86°
10 %	10 cm per meter horizontally	5,71°
30 %	30 cm per horizontal meter	16,70°
50 %	50 cm per meter horizontally	26,57°
100 %	1 m per horizontal meter	45,00°

Three mistakes often occur: dividing centimeters by meters, putting the inclined length as the denominator instead of the horizontal distance, and reading a result in degrees as radians given by the calculator. In the latter case, multiply by 180 and divide by π, once only.

Roof slope application: the inclination of a slope

For a two-slope roof, the slope is read on a cut perpendicular to the eaves. The elevation is the height of the slope, between the gutter, its low edge, and the eaves, its high line, and not the total height of the building. The horizontal distance is the projection of the slope, from the gutter to the eaves: it is only half the width of the roof, including the gutter overflow, if the eaves are centered. Always measure in the direction of the steeper slope, because a rafter has not the same inclination as the slopes it separates.

On a slope, for example, the rise is 2.50 m for a horizontal projection of 4.00 m: 100 × 2.50 ÷ 4.00 = 62.5%, i.e., arctan(0.625) = 32.01°. A roof pitch application does this calculation for you, but it does not choose the correct angles for you. As explained by the VELUX guide on the roof pitch, Everything starts from properly measured horizontal height and distance.

The obtained slope is then used to verify that the planned coverage suits it. Minimum laying slope It depends on the material, the geographical area and the exposure of the site, as is recalled wienerberger for tiled roofs ; Manufacturers' drawings also include the horizontal projection of the eaves. Therefore, no single value applies to all roofs.

Calculating the roof slope: the actual length of the slope

The slope is the inclined length of the slope, from the gutter to the eaves. It is derived from the gradient and the projection: slope = √(gradient² + projection²) For the previous slope, √(2.50² + 4.00²) = 4.72 m. Do not confuse the two lengths: 4.00 m is a horizontal distance, readable on a roof plan viewed from above, whereas 4.72 m is the inclined length, revealed by a section, a gable façade or calculation.

For a rectangular roof, the calculated pitch allows us to move on to the actual surface: pitch × length of the roof, measured parallel to the eaves. A single elevation view is not enough, because of the height difference. For a complete two- or four-slope roof, our page on the calculation of the roof surface details the method.

Do you have the PDF of the cut? Measure the slope of the slope directly on the plan with MeasureAll: a complete file is provided.

Calculation of terrain slope: elevation, distance and methods of measurement

For a terrain, there must be two points A and B, their elevation in the same reference system and the horizontal distance separating them:

Slope (%) = 100 × (altitude of B − altitude of A) ÷ horizontal distance

The sign indicates the direction: positive when going up, negative when going down from A to B. If you have only measured the inclined length L, first find the horizontal distance: D = √(L² − dénivelé²).

Let’s take a rectangular plot of 20 × 30 m, with a vertical difference of 2.50 m between two opposite corners. The horizontal diagonal measures √(20² + 30²) = 36.06 m, giving an average slope of 100 × 2.50 ÷ 36.06 = 6.93%. This figure remains an average over a single direction: the terrain may be steeper elsewhere, or present a break in the slope. For useful results, measure several directions and note the direction of each measurement.

Method	Principle	Limit
Rule and level in bubble	Horizontal rule held, vertical gap measured to the ground, by sections	Local survey; the elevations of each section are added together
Optical or laser level, with sight	Reading the mire, a graduated rule, in A then in B, without moving the device	Stable station and common reference indispensable
Topographic plan PDF	Heights shown on the plan, distance measured between the points shown	No points marked or level curves, no elevation gain
Altimetry profile online	Profile traced on a map between two points	Indicative result, according to the altimetric data used
Left of the surveyor	Contour points, slope breaks and terrain profiles	Mission and precision to be specified in the quote

With an optical level, or a laser whose beam remains horizontal, the ruler holds in a line: altitude of B = altitude of A + reading in A − reading in B, from the same station, that is to say without moving the device. If you read 0.725 m on the aiming device in A and 1.465 m in B, at a horizontal distance of 12.40 m, the elevation is 0.725 − 1.465 = −0.74 m. The slope is therefore 100 × (−0.74) ÷ 12.40 = −5.97%: the terrain slopes from A to B. Leica NA2 level manual, An optical level describes this reading principle.

Online, the former Géoportail has given way to cartes.gouv.fr. Its tool for altimetric profile gives an overview of the relief, but the result depends on the given altimetric value used, of its resolution and precision. For an altitude read on a map at 1/25,000, the IGN geodetic FAQ indicates a precision of only a few meters: at a distance of 20 m, an error of one meter in elevation changes the slope by 5 percentage points. Finally, the’angle measuring tool The map gives an angle relative to the North, an azimuth, and not a slope.

A slope alone does not say whether a project is feasible. Operational urban planning certificate Indicates whether the proposed operation is feasible in the field. The accessibility thresholds depend on the applicable text, such as the current slope of 5% for a new ERP access path. For a garage descent, no universal maximum is established, and the average can mask a steep part: a 1.80 m elevation difference over 12 m gives an average of 15%. If the first 4 meters are limited to 5%, they only recover 0.20 m: there remains 1.60 m over 8 m, or 20%, not counting the connections at the top and bottom.

Calculation of stair slope: step height and step angle

For a straight line with regular steps, the slope is calculated using two quantities: the step height h and the angle g, that is, the horizontal distance between two successive steps. Angle = arctan(h ÷ g) and slope = 100 × h ÷ g. The angle is not necessarily the total depth of the step: measure it between the steps.

For example, a staircase has 2.72 m of steps in 16 steps, i.e., h = 2.72 ÷ 16 = 0.17 m. With a step height of 0.28 m, the angle is arctan(0.17 ÷ 0.28) = 31.26°, or a slope of 60.71%.

The slope of a staircase says nothing about comfort or conformity. Comfort is appreciated with the Blondel relationship, 2h + g, which here is 2 × 0.17 + 0.28 = 0.62 m. The recommended ranges vary according to the sources: from 60 to 66 cm in the Design guide from the INRS for workplaces, from 62 to 65 cm in the CAUE sheet for Occitanie on the masonry staircase. The INRS reminds that this relationship alone does not guarantee a safe staircase.

The regulations depend on the building. For a staircase open to the public in a new ERP, the’Order of April 20, 2017 Limits the step height to 16 cm and imposes a step height of at least 28 cm. A staircase with a step height of 17 cm and a step width of 29 cm gives a step length of 2 × 17 + 29 = 63 cm, within the ranges cited, and an angle of 30.38°, i.e., arctan(17 ÷ 29). However, its step height exceeds this limit. An application calculates the geometry; it does not validate a staircase.

Calculator, inclinometer or PDF plan: which application to use to calculate a slope?

Before choosing an application to calculate a slope, identify the data you have at your disposal: the right tool depends on it.

Tool	What he gives	What he asks for	Limit
Online calculator	The slope starting from two lengths	Gradient and horizontal distance already known	Does not measure anything itself
Electronic level or inclinometer	The angle of the support on which it rests	Physical access to the surface	Local measurement, based on support and calibration
Smartphone level	An indicative inclination	The phone placed on the surface	Measure presented as approximate by Apple
Measured on a PDF plan	Lengths, angles and slopes shown on the drawing	A plan on a scale, with a reliable rating	Depends on the plan and the calibration

An electronic level like the GIM 120 from Bosch displays the inclination in degrees, percentages or mm/m, with a declared accuracy of ±0.2° between 1° and 89°: this is the characteristic of this instrument, not that of a telephone. Apple also presents the iPhone level As an approximate measure. A photo of the roof does not replace a drawing: perspective distorts angles and lengths. For a building whose plans are available, or a project that has yet to be drawn, the PDF measurement gives the dimensions without access to the site.

MeasureAll: measuring a slope on a PDF plan

MeasureAll is a PDF plan measurement app for iPhone, iPad and Android that also serves as an application for calculating a slope from a drawing. It measures on the drawing, not on the actual surface. The path follows the steps below, which can be adapted to the document you have at hand:

1. Import and calibrate. You import the plot, elevation, or topographic plan, calibrate it to a known elevation, and then check a second elevation.
2. Lift the lengths. On a map, you measure the elevation and the horizontal projection of the slope; for a flight, the walking height and the ridge, or the difference between two walking noses; on a topographic map, the distance between two points on the side.
3. Measure the angle. Using the angle tool, mark on the drawing the angle between the slope, or the line of the walking noses, and the horizontal line, provided that the section has the same scale in both horizontal and vertical directions.
4. Apply the slope. For a roof, the specified slope is used to calculate the inclined surface of the slope.
5. Classify and export. You name the measures, you group them by project, and you export them in a spreadsheet-readable format.

On a long profile, the horizontal and vertical scales often differ: the drawn angle is not then the actual slope. Instead, read the quoted altitudes and distances, then apply the formula. Once the plans are loaded, the survey also works offline. The accuracy depends on the plan, its scale and the care taken in the calibration: always check a second reference point before relying on the result.

Measure the slopes of your PDF plan: a complete file is provided. MeasureAll is available on the’App Store for iPhone and iPad, and on Google Play for Android smartphones and tablets. After this first subscription, the subscription is purchased in the app.

Application for calculating a slope: what you need to remember

• Slope (%) = 100 × elevation ÷ horizontal distance, and angle = arctan(slope ÷ 100): a slope of 100% makes 45°, not 90°.
• For a roof, measure the height of the slope and its projection on a section; the slope is equal to √(slope² + projection²).
• For a terrain, start with two altitudes and a horizontal distance, and remember that the slope changes depending on the direction.
• For a staircase, calculate the height of the steps and the landings: neither the angle nor the Blondel ratio are valid for compliance.
• Choose the tool according to your data: calculator, instrument on site or measured on a PDF plan.

For a roofing, landscaping or general contracting company, MeasureAll offers annual subscriptions for teams, with company code activation, a single invoice and transferable licenses. Equip your construction teams: request a company quote.