Distance Formula Calculator
Calculate the distance between two points on a 2D plane
Point 1 (x₁, y₁)
Point 2 (x₂, y₂)
Distance (d)
Exact details below
Exact Value Format
Deltas (Δx, Δy)
Geometry Triangle
Illustration representation of right triangle formed by the two points.
Step-by-Step Solution
Distance Formula Properties
Basic Rules
- • Distance is always positive or zero.
- • If distance is zero, the two points are exactly the same.
- • The order of the points doesn't matter (Point 1 to Point 2 is the same distance as Point 2 to Point 1).
Distance vs. Midpoint
- • Distance: Measures the length of the segment connecting the points. Formula uses differences.
- • Midpoint: Finds the exact middle position between the points. Formula uses averages.
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About Distance Formula Calculator
What is the Distance Formula?
The distance formula is a fundamental algebraic tool used to measure the straight-line distance between two points on a two-dimensional Cartesian coordinate system. It is derived directly from the Pythagorean theorem.
How to Use This Calculator
- Point 1: Enter the x and y coordinates for your first point (x₁, y₁).
- Point 2: Enter the x and y coordinates for your second point (x₂, y₂).
- Result: The calculator instantly computes the exact distance, displaying the final number as well as showing the step-by-step mathematical breakdown.
Understanding the Distance Formula
The Formula
The standard distance formula for two points (x₁, y₁) and (x₂, y₂) is:
d = √[ (x₂ - x₁)² + (y₂ - y₁)² ]
Where:
- d is the distance between the two points
- (x₁, y₁) are the coordinates of the first point
- (x₂, y₂) are the coordinates of the second point
How It Works (Example)
Let's find the distance between point A (3, 2) and point B (7, 5):
- Identify coordinates: x₁=3, y₁=2, x₂=7, y₂=5
- Substitute into formula: d = √[ (7 - 3)² + (5 - 2)² ]
- Calculate differences: d = √[ 4² + 3² ]
- Square the differences: d = √[ 16 + 9 ]
- Add the squares: d = √[ 25 ]
- Take the square root: d = 5
The distance between (3, 2) and (7, 5) is exactly 5 units.
Frequently Asked Questions
Does it matter which point is Point 1 and which is Point 2?
No, it does not matter. Because the differences (x₂ - x₁) and (y₂ - y₁) are squared, any negative result becomes positive. So, √[ (x₁ - x₂)² + (y₁ - y₂)² ] will give the exact same result.
Can the distance ever be negative?
No. Distance is a measure of length, which is a scalar quantity and is always positive or zero. The square root symbol in the formula signifies the principal (positive) square root.
What if the distance is exactly zero?
If the distance is zero, it means that Point 1 and Point 2 are at the exact same location; they share the same x and y coordinates.
Is this the same as the Pythagorean theorem?
Yes, the distance formula is just an application of the Pythagorean theorem (a² + b² = c²) plotted on a coordinate plane. The difference in x-values represents one leg of a right triangle, the difference in y-values represents the other leg, and the distance between the points represents the hypotenuse.