Angle & Slope Converter

Formula notes

• The basic trigonometric relationship is tan(theta) = rise / run.
• Decimal slope = vertical rise / horizontal run.
• Slope percent = decimal slope x 100.
• Angle in radians = atan(decimal slope).
• Angle in degrees = atan(decimal slope) x 180 / pi.
• For a slope ratio written as 1V:xH, decimal slope = 1 / x.
• For a slope ratio written as 1V:xH, horizontal component x = 1 / decimal slope.
• Degrees and radians describe the same physical angle using different angular units.
• Slope percentage and slope ratio are dimensionless, so the same result is obtained whether rise and run are measured in millimetres, metres, feet, inches, or any other consistent length unit.
• Worked example: for an angle of 5 degrees, decimal slope = tan(5 degrees) = approximately 0.087489. The slope is approximately 8.7489 percent, the angle is approximately 0.087266 radians, and the equivalent ratio is approximately 1V:11.4301H.
• QA test 1: input 5 degrees => expected radians approximately 0.087266; slope approximately 8.7489 percent; ratio approximately 1V:11.4301H.
• QA test 2: input 10 percent slope => expected decimal slope 0.100000; angle approximately 5.7106 degrees; radians approximately 0.099669; ratio exactly 1V:10H.
• QA test 3: input ratio 1V:20H => expected decimal slope 0.050000; slope 5.0000 percent; angle approximately 2.8624 degrees; radians approximately 0.049958.
• QA test 4: input 45 degrees => expected radians approximately 0.785398; decimal slope 1.000000; slope 100.0000 percent; ratio 1V:1H.
• QA test 5: input 30 degrees => expected radians approximately 0.523599; decimal slope approximately 0.577350; slope approximately 57.7350 percent; ratio approximately 1V:1.7321H.
• QA test 6: input ratio 1V:2H => expected decimal slope 0.500000; slope 50.0000 percent; angle approximately 26.5651 degrees; radians approximately 0.463648.
• QA test 7: input 0.1 radians => expected angle approximately 5.7296 degrees; decimal slope approximately 0.100335; slope approximately 10.0335 percent; ratio approximately 1V:9.9666H.

Assumptions

• Slope is treated as vertical rise divided by horizontal run.
• Slope ratio is expressed in the civil engineering convention 1V:xH.
• The calculator assumes positive upward or downward slope magnitude and reports the magnitude only.
• Angles must be greater than zero and less than 90 degrees because a vertical slope approaches an infinite slope percentage and a horizontal slope would produce an infinite 1V:xH ratio.
• For slope percentage input, 100 percent means a rise equal to the horizontal run and therefore corresponds to an angle of 45 degrees.
• A 50 percent slope corresponds to a decimal slope of 0.5 and a ratio of 1V:2H.
• A ratio of 1V:10H corresponds to a decimal slope of 0.1 and a slope of 10 percent.
• SI and US customary length systems do not require separate conversion because slope is dimensionless when rise and run use consistent length units.
• The calculator does not automatically interpret alternative ratio notation such as H:V. Enter the horizontal component using the stated 1V:xH convention.
• The calculations use ideal geometric relationships and do not account for surveying measurement uncertainty, construction tolerances, road crossfall conventions, drainage criteria, accessibility requirements, or code-specific maximum slopes.

References

• Basic right-triangle trigonometry: tan(theta) = opposite / adjacent.
• Standard civil engineering slope and gradient conventions.
• Applicable project drawings, surveying procedures, highway geometric design criteria, drainage requirements, accessibility requirements, and local engineering standards.