Distance from a Point to a Line Calculator
Find the shortest distance from a point (x₀, y₀) to the line Ax + By + C = 0, with the exact value, the foot of the perpendicular and whether the point lies on the line.
How it works
d = |A·x₀ + B·y₀ + C| / √(A² + B²) · foot of the perpendicular: (x₀ − A·k, y₀ − B·k), k = (A·x₀ + B·y₀ + C)/(A² + B²)
The shortest path from a point to a line is the perpendicular segment. Plugging the point into Ax + By + C gives a value proportional to its signed distance, and dividing by the length of the normal vector (A, B) turns it into the actual distance. If the line is in slope-intercept form y = mx + b, rewrite it as mx − y + b = 0, so A = m, B = −1 and C = b.
Worked example
For the line 3x + 4y − 10 = 0 and the point (5, 6): 3·5 + 4·6 − 10 = 29 and √(3² + 4²) = 5, so d = 29/5 = 5.8. The closest point on the line is (5 − 3·29/25, 6 − 4·29/25) = (1.52, 1.36).
Sources: Distance from a point to a line — Wikipedia · Point-Line Distance — Wolfram MathWorld
Frequently asked questions
How do I use y = mx + b with this formula?
Move everything to one side: mx − y + b = 0. Then A = m, B = −1, C = b. For y = 2x + 1 enter A = 2, B = −1, C = 1.
What if the line is vertical or horizontal?
A vertical line x = k is 1·x + 0·y − k = 0, so the distance is |x₀ − k|; a horizontal line y = k gives |y₀ − k|. The general formula handles both.
What does a distance of 0 mean?
The point satisfies the equation, so it lies on the line.
How do I find the distance between two parallel lines?
Write both as Ax + By + C₁ = 0 and Ax + By + C₂ = 0 with the same A and B; the distance is |C₁ − C₂| / √(A² + B²). Or pick any point on one line and use this calculator with the other.
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This calculator is for educational purposes. Double-check important results before acting on them.