Distance Formula Calculator

Distance Formula

The distance formula calculates the straight-line distance between two points on a coordinate plane:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Pythagorean Theorem Connection

The distance formula is derived from the Pythagorean theorem. When you connect two points on a coordinate plane, you create the hypotenuse of a right triangle. The horizontal and vertical distances form the legs:

Horizontal leg: |x₂ - x₁|
Vertical leg: |y₂ - y₁|
Hypotenuse (distance): √[(x₂ - x₁)² + (y₂ - y₁)²]

Real-World Applications

GPS navigation and mapping systems
Computer graphics and game development
Physics calculations for displacement
Architecture and engineering design
Astronomy for calculating distances between celestial objects

Frequently Asked Questions

What is the distance formula?

The distance formula is d = √[(x₂-x₁)² + (y₂-y₁)²], which calculates the straight-line distance between two points (x₁, y₁) and (x₂, y₂) on a coordinate plane.

How is the distance formula derived?

The distance formula comes from the Pythagorean theorem (a² + b² = c²). The horizontal and vertical distances between two points form the legs of a right triangle, and the distance is the hypotenuse.

Can the distance formula be used in 3D space?

Yes! For 3D space, the formula extends to d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²], adding the third dimension.

Does the order of points matter?

No, the distance formula gives the same result regardless of which point you label as point 1 or point 2, because we square the differences.

What if one or both coordinates are negative?

The formula works with negative coordinates. The squaring step ensures all values become positive, so the distance is always a positive number.

How do I find the distance from the origin?

To find distance from the origin (0, 0), simply use d = √(x² + y²) where (x, y) is your point.

What units does the distance formula use?

The distance is in the same units as your coordinate system. If your axes are in meters, the distance is in meters. On a pure coordinate plane, it's in "units."

Can I use the distance formula for diagonal measurements?

Absolutely! The distance formula is perfect for finding diagonal measurements in rectangles, squares, or any geometric figure on a coordinate plane.