Calcand

Distance Calculator

Distance Formula

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

The distance formula is derived from the Pythagorean theorem. It calculates the straight-line distance between two points in a 2D coordinate plane.

Quick Examples

(0,0) to (3,4)
Distance = 5
(1,1) to (4,5)
Distance = 5
(0,0) to (1,1)
Distance = √2 ≈ 1.414

Related Calculators

Midpoint CalculatorSlope CalculatorPythagorean Theorem

Understanding the Distance Formula

What is Distance?

In coordinate geometry, the distance between two points is the length of the straight line connecting them. This is also called Euclidean distance, named after the ancient Greek mathematician Euclid.

Distance Formula

d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Where (x₁, y₁) and (x₂, y₂) are the two points

Derivation from Pythagorean Theorem

The distance formula comes directly from the Pythagorean theorem. Given two points, we can form a right triangle where:

  • The horizontal leg has length |x₂ - x₁|
  • The vertical leg has length |y₂ - y₁|
  • The hypotenuse is the distance we want to find

Step-by-Step Example

Points: (1, 2) and (4, 6)
Δx = 4 - 1 = 3
Δy = 6 - 2 = 4
d = √(3² + 4²) = √(9 + 16) = √25 = 5

Real-World Applications

Navigation & GPS

  • Calculate straight-line distance between locations
  • Route planning and optimization
  • Geofencing applications

Computer Graphics

  • Collision detection in games
  • Image processing
  • 3D modeling and rendering

Important Note

The distance formula gives the straight-line distance. For real-world navigation (driving, walking), actual travel distance may be longer due to roads, obstacles, and terrain. For spherical surfaces like Earth, use the Haversine formula instead.

Frequently Asked Questions

What is the distance formula?

The distance formula is d = √[(x₂ - x₁)² + (y₂ - y₁)²]. It calculates the straight-line (Euclidean) distance between two points in a 2D plane. This formula is derived from the Pythagorean theorem, where the distance is the hypotenuse of a right triangle formed by the horizontal and vertical distances.

How is the distance formula related to the Pythagorean theorem?

The distance formula is a direct application of the Pythagorean theorem (a² + b² = c²). The horizontal distance (x₂ - x₁) is one leg, the vertical distance (y₂ - y₁) is the other leg, and the distance between points is the hypotenuse. Solving for c gives us the distance formula.

Can I use this formula for 3D distance?

For 3D distance, extend the formula to d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]. Simply add the squared difference of the z-coordinates under the square root. The principle remains the same.

What units does the distance calculator use?

The calculator works with whatever units you input. If your coordinates are in meters, the distance is in meters. If they're in feet, the distance is in feet. Just ensure both points use the same unit system.