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Triangle Slope Calculator

Instantly calculate slope percentage and angle in degrees from rise and run — ideal for road design, roofing, ramp planning, and landscaping projects.

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Inputs

Rise (Vertical Distance)

Run (Horizontal Distance)

Output Type

Slope Result

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Slope Result—

The formula

How the
result is
computed.

How the Triangle Slope Calculator Works

The triangle slope calculator determines slope steepness using two industry-standard measurements: percent grade and angle in degrees. Both measurements derive from right triangle geometry, where the rise is the vertical leg and the run is the horizontal leg of the triangle.

The Core Formulas

Two formulas power this calculator:

Percent Slope: Slope % = (rise / run) x 100
Angle of Slope: θ = arctan(rise / run)

The percent slope formula expresses steepness as a ratio of vertical change to horizontal change, scaled to a percentage. A rise of 3 feet over a run of 100 feet produces a 3% slope. The angle formula applies the inverse tangent (arctan) function to that same ratio, converting it to degrees. According to the U.S. Geological Survey, percent slope and angle of slope are the two standard methods for expressing terrain steepness in topographic and engineering contexts.

Understanding Rise and Run

The rise is the vertical distance — the elevation change between two points. The run is the horizontal distance — the flat-plane measurement between those same two points. Together, rise and run form the two legs of a right triangle, with the slope line serving as the hypotenuse. Portland Community College defines slope as the ratio of vertical change to horizontal change and identifies it as foundational to linear algebra and coordinate geometry. See: Let's Learn Slope of a Line.

Percent Slope vs. Angle in Degrees

Percent slope and angle in degrees describe the same steepness using different scales:

A 100% slope equals exactly 45°, because arctan(1) = 45°.
A 10% slope equals approximately 5.71°.
A 50% slope equals approximately 26.57°.
A 200% slope equals approximately 63.43°.

Percent grade dominates civil engineering, road design, and drainage because it directly quantifies rise per 100 units of run. Angle in degrees is more common in structural engineering, trigonometry, and navigation.

Right Triangle Trigonometry

Every slope calculation is a right triangle problem. The tangent of an angle in a right triangle equals the ratio of the opposite side to the adjacent side — precisely the rise-to-run relationship. As Clark University's trigonometry reference explains: tan(θ) = opposite / adjacent = rise / run, therefore θ = arctan(rise / run). See: Right Triangles — Dave's Short Trig Course. The actual slope distance (hypotenuse) can also be derived using the Pythagorean theorem: hypotenuse = sqrt(rise^2 + run^2).

Real-World Applications

Triangle slope calculations appear across many professional disciplines:

Road and Highway Design: Maximum grades on U.S. highways typically fall between 6% and 8%. The Texas Department of Transportation uses percent slope to model gutter flow and drainage performance. See: TxDOT Gutter and Inlet Equations.
Roof Pitch: A 4:12 roof pitch equals a 33.3% slope and approximately 18.4°. A 6:12 pitch equals 50% and approximately 26.6°.
Accessibility Ramps: ADA-compliant ramps require a maximum slope of 1:12 (8.33%), equivalent to 4.76°.
Landscaping and Drainage: Proper surface drainage requires at least a 2% slope away from building foundations to prevent water infiltration.
Hiking and Trail Design: Sustainable trail grades typically stay below 15% to minimize erosion and maintain hiker safety.

Step-by-Step Example

Given a rise of 6 feet and a run of 24 feet:

Percent slope = (6 / 24) x 100 = 25%
Angle = arctan(6 / 24) = arctan(0.25) = 14.04°

This result describes a moderate grade — steep enough to affect vehicle fuel economy and surface drainage, but within the acceptable range for most residential driveways and trail systems.

Why Use This Calculator

Manual arctan calculations require a scientific calculator and careful unit consistency. The triangle slope calculator accepts rise and run in any consistent unit — feet, meters, or inches — and instantly returns both slope percentage and angle in degrees, eliminating unit-conversion errors and saving time for engineers, contractors, landscapers, and students alike.

Reference

Frequently asked questions

What is a triangle slope calculator?

A triangle slope calculator computes slope steepness from two inputs: rise (vertical distance) and run (horizontal distance). It outputs either percent grade using the formula (rise / run) x 100, or angle in degrees using arctan(rise / run). Common uses include road grading, roof pitch calculations, ramp design, stormwater drainage planning, and land surveying.

What is the difference between percent slope and angle of slope?

Percent slope expresses steepness as a percentage: a 3-foot rise over a 100-foot run equals 3%. Angle of slope uses the arctan function to convert that ratio to degrees. A 100% slope equals exactly 45°, because arctan(1) = 45°. Engineers prefer percent grade for road and drainage design, while trigonometry and structural engineering more commonly use degree measurements.

How do you calculate percent slope from rise and run?

Divide the rise by the run, then multiply by 100. For example, a 5-meter rise over a 20-meter run equals (5 / 20) x 100 = 25% slope. Both values must use the same unit of measurement — mixing feet and meters produces incorrect results. This formula works for any consistent unit including inches, feet, meters, or yards.

What slope percentage is considered steep for roads and driveways?

For roads, grades above 6 to 8% are considered steep; U.S. highway design standards typically cap grades at 8%. Residential driveways above 15% can present traction challenges in wet or icy conditions. Ski slopes range from roughly 25% on beginner runs to over 100% on expert terrain. For drainage systems, a 0.5 to 2% slope is generally sufficient to move water effectively.

How is slope used in drainage and stormwater engineering?

Civil engineers use percent slope to calculate flow velocity in gutters, channels, and storm drains. Steeper slopes increase flow velocity and reduce ponding risk, but excessive slope causes erosion and structural damage to channels. The Texas Department of Transportation's hydraulic design manual applies slope values directly in gutter and inlet flow equations to ensure safe drainage at roadways and intersections.

How do you convert a slope angle in degrees to percent grade?

Multiply the tangent of the angle by 100. For a 15-degree slope: tan(15°) is approximately 0.2679, so the percent grade is approximately 26.79%. To convert percent grade back to degrees, take the arctan of the decimal form: arctan(0.2679) is approximately 15°. Scientific calculators include an arctan key for this conversion; the triangle slope calculator performs both conversions automatically.