Point to Line Distance Calculator

Math Geometry • Coordinate Geometry

Written by STEM Calculators Team Published January 23, 2026 Updated February 24, 2026

Find the shortest (perpendicular) distance from a point to a line. Works for vertical/horizontal lines too. The graph is interactive: drag to pan, wheel/trackpad to zoom, pinch on touch.

Inputs accept 1e-3, pi, e, sqrt(2), sin(), cos(), tan(), ln(), log(), abs(). Use * for multiplication.

Inputs

Line input mode:

Point

\(x_0\):

\(y_0\):

Line

\(a\):

\(b\):

\(c\):

Slope \(m\):

Intercept \(k\):

\(x_1\):

\(y_1\):

\(x_2\):

\(y_2\):

Tip: In general form, scaling \((a,b,c)\) by a nonzero constant does not change the distance.

Graph options

Axis units label: Show grid: Show tick labels:

Graph uses square units: 1 unit in \(x\) equals 1 unit in \(y\).

Ready

Choose a mode and click Calculate.

Point-to-line distance (pan/zoom enabled)

Drag to pan • wheel/trackpad to zoom • pinch on touch • “Reset view” fits the geometry.

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Theory

Distance from a point to a line (general form)

Let the line be written in general form:

\[ ax+by+c=0, \]

and let the point be \(P(x_0,y_0)\). The shortest distance from the point to the line is the perpendicular distance:

\[ d=\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}. \]

The pair \((a,b)\) is a normal vector to the line, so \(\sqrt{a^2+b^2}\) is its length. If you multiply \(a,b,c\) by the same nonzero constant, the distance stays the same.

The line is valid only if \(a\) and \(b\) are not both zero.

Foot of the perpendicular (closest point on the line)

The closest point \(H(x_H,y_H)\) on the line to \(P(x_0,y_0)\) is called the foot of the perpendicular. Define

\[ t=\frac{ax_0+by_0+c}{a^2+b^2}. \]

Then

\[ H=\bigl(x_0-at,\;y_0-bt\bigr). \]

The segment \(\overline{PH}\) is perpendicular to the line, and its length is exactly \(d\).

Common line forms and conversion

Slope–intercept: \(y=mx+k \;\Rightarrow\; mx-y+k=0\).
Two points: through \((x_1,y_1)\) and \((x_2,y_2)\), \[ (y_1-y_2)x+(x_2-x_1)y+(x_1y_2-x_2y_1)=0. \]

Once you have \(a,b,c\), the distance formula applies directly.

Worked example

Point \(P(1,1)\), line \(2x-y+3=0\).

\[ d=\frac{|2(1)+(-1)(1)+3|}{\sqrt{2^2+(-1)^2}} =\frac{|4|}{\sqrt{5}} \approx 1.789. \]

If you expected a different number, double-check the line coefficients and signs in the general form.

Frequently Asked Questions

What is the formula for the distance from a point to a line?

For a line ax + by + c = 0 and a point (x0, y0), the perpendicular distance is d = |a x0 + b y0 + c| / sqrt(a^2 + b^2). The line is valid only if a and b are not both zero.

Why does multiplying (a, b, c) by a constant not change the distance?

Scaling a, b, and c by the same nonzero constant represents the same geometric line. The numerator and denominator scale by the same factor in the distance formula, so the ratio stays the same.

How do you convert slope-intercept form to general form for the distance formula?

Starting from y = m x + k, move all terms to one side to get m x - y + k = 0. Then use a = m, b = -1, and c = k in the distance formula.

Does this work for vertical or horizontal lines?

Yes. Vertical and horizontal lines are handled by the general-form distance formula, so the calculator still returns the perpendicular distance without needing a slope.

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Frequently Asked Questions

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