Perpendicular Slope CalculatorFree online math calculator — instant results, no sign-up
Find the slope of a line perpendicular to a given nonhorizontal line.
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About the Perpendicular Slope Calculator
The Perpendicular Slope Calculator finds the slope of a line that meets a given line at a right angle. Enter a finite, nonzero original slope, and the tool returns its negative reciprocal. With the default original slope of 2, the perpendicular slope is -0.5.
- Original slope
- The original slope is the line's change in y divided by its change in x, expressed in y-units per x-unit and dimensionless when both axes use the same units.
Formula and method
m_perp = -1 / m; with m = 2, m_perp = -1 / 2 = -0.5.
- mThe finite, nonzero slope of the original line.
- m_perpThe slope of any line perpendicular to the original line.
Divide -1 by the original slope to obtain its negative reciprocal. For the default input, dividing -1 by 2 gives -0.5. The result can be checked using m × m_perp = -1: 2 × (-0.5) = -1.
Worked example
These are the calculator's starting values and the results it actually produces for them.
Inputs
- Original slope2
Results
- Result-0.5
How to use the Perpendicular Slope Calculator
- 1Identify the original line's slope from its equation or from its rise divided by its run.
- 2Enter that value in Original slope; leave the default value of 2 to follow the worked example.
- 3Check that the slope is finite and not zero before calculating.
- 4Read the Result row: for an original slope of 2, it shows -0.5.
- 5Confirm perpendicularity by checking that the original slope multiplied by the result equals -1.
Reading your result
The Result row gives the perpendicular line's slope, not its intercept or a complete line equation. With the default input of 2, Result = -0.5 means the perpendicular line falls by 0.5 y-units for each x-unit moved to the right. Any line with this slope is perpendicular to the original line under standard Cartesian geometry.
Limitations
The formula requires a finite, nonzero slope. A horizontal original line has slope zero, so its perpendicular is vertical and has undefined slope rather than a finite numerical result. A vertical original line also has undefined slope and cannot be supplied as a finite slope input; its perpendicular is horizontal. The rule assumes perpendicular Cartesian axes with consistent coordinate units; unequal display scales can distort the apparent angle. A slope alone does not determine a unique line, and displayed decimals may be rounded.
Common questions
Why does an original slope of 2 give -0.5 rather than -2?+
Perpendicular slopes are negative reciprocals, not just opposite signs. Taking the reciprocal of 2 and changing its sign gives -1 / 2 = -0.5.
What happens if the original slope is zero?+
A zero slope describes a horizontal line. Its perpendicular is vertical, so its slope is undefined and the formula -1 / m cannot return a finite result.
Can I use this calculator if the original line is vertical?+
Not through a finite numerical slope input, because a vertical line's slope is undefined. Its perpendicular line is horizontal and has slope zero.
Does the result give the equation of the perpendicular line?+
No. The result gives only its slope. For the default result, the family of perpendicular lines is y = -0.5x + b; a point on the desired line is needed to determine the intercept b.
Does the calculator save my information?+
Calculations run in your browser. While signed in, inputs and results are automatically recorded in account history; saving a result creates a separate saved record.
Further reading
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Results are estimates for informational purposes and are not financial, medical or legal advice.