Case 1
- Coefficient a
- 3
- Coefficient b
- 4
- Constant c
- -5
- Point x-coordinate (x₀)
- 0
- Point y-coordinate (y₀)
- 0
1.000000
Open with these valuesDistance from a Point to a Line
1.000000
Result: 1.000000Write the line as a·x + b·y + c = 0, put the point into the left-hand side, take the absolute value and divide by √(a² + b²). For 3x + 4y − 5 = 0 and the origin that is 5 ÷ 5 = 1. A line given as y = mx + k becomes a = m, b = −1, c = k.
The numbers at a glance
Held fixed: Coefficient a 3.0000, Coefficient b 4.0000, Constant c -5.0000, Point x-coordinate (x₀) 0.0000.
| Point y-coordinate (y₀) | Result |
|---|---|
| -40,000.0000 | 32,001.000000 |
| -20,000.0000 | 16,001.000000 |
| 0.0000Your value | 1.000000 |
| 20,000.0000 | 15,999.000000 |
| 40,000.0000 | 31,999.000000 |
Worked examples
Case 2
- Coefficient a
- 5
- Coefficient b
- -12
- Constant c
- 2
- Point x-coordinate (x₀)
- 1
- Point y-coordinate (y₀)
- 1
0.384615
Open with these valuesCase 3
- Coefficient a
- 1
- Coefficient b
- 2
- Constant c
- 2
- Point x-coordinate (x₀)
- -3
- Point y-coordinate (y₀)
- -4
4.024922
Open with these valuesHow it's calculated
d = |a·x₀ + b·y₀ + c| ÷ √(a² + b²)
- StepBring the line into the form a·x + b·y + c = 0 and enter a, b and c.
- StepEnter the coordinates of the point.
- ResultThe point goes into the left-hand side; the absolute value is divided by √(a² + b²).
Reference table
| a, b, c, x₀, y₀ | Numerator ÷ denominator | Distance |
|---|---|---|
| 3, 4, -5, 3, -1 | 0 ÷ 5 | 0.000000 |
| 1, 1, 0, 3, -3 | 0 ÷ √2 | 0.000000 |
| 5, -12, 2, 1, 1 | 5 ÷ 13 | 0.384615 |
| 3, 4, -5, 0, 0 | 5 ÷ 5 | 1.000000 |
| 1, 2, 2, -3, -4 | 9 ÷ √5 | 4.024922 |
| 0, 1, -3, 2, 7 | 4 ÷ 1 | 4.000000 |
Questions
How do I calculate the distance from a point to a line?
Write the line in general form a·x + b·y + c = 0, then use d = |a·x₀ + b·y₀ + c| ÷ √(a² + b²). For the line 3x + 4y − 5 = 0 and the origin that is |−5| ÷ √25 = 1.
My line is y = mx + k — how do I use this calculator?
Move everything to one side: m·x − y + k = 0. So a = m, b = −1 and c = k. The line y = 2x + 1 becomes a = 2, b = −1, c = 1.
What does perpendicular distance mean here?
It is the shortest straight-line gap between the point and the line, measured along a segment that meets the line at a right angle. Every other path from the point to the line is longer.
Why does the formula use absolute value?
Because a distance is never negative. The expression a·x₀ + b·y₀ + c comes out positive or negative depending on which side of the line the point lies, so the absolute value strips the sign. Check that sign before taking it if you need the side.
What happens if a and b are both zero?
Then a·x + b·y + c = 0 describes no line at all and the denominator √(a² + b²) is zero, so the distance is undefined. At least one of a and b must be non-zero, and the calculator shows no result otherwise.
Sources and last check
Information, not professional advice.