How this instrument works
The Distance Formula finds the straight-line distance between two points on a coordinate plane by treating the horizontal gap between them and the vertical gap between them as the two legs of a right triangle, then applying the Pythagorean theorem: d = √((x₂−x₁)² + (y₂−y₁)²). The distance itself plays the role of the triangle's hypotenuse, the longest side, connecting the two points directly rather than by going across and then up.
This is genuinely just the Pythagorean theorem restated in coordinate language: the horizontal leg has length |x₂−x₁|, the vertical leg has length |y₂−y₁|, and squaring each of those (which erases any negative sign automatically) before adding and taking a square root gives the direct distance exactly as a² + b² = c² would for any right triangle.
The formula works identically no matter which point is labeled 'first' and which is labeled 'second' — swapping the two points flips the sign of both differences, but squaring erases that sign either way, so the resulting distance never changes. It also correctly returns zero when the two points coincide, since both differences vanish.
- Enter the first point's coordinates into the Point 1: x and Point 1: y fields.
- Enter the second point's coordinates into the Point 2: x and Point 2: y fields.
- Read Distance: the sheet applies the Distance Formula directly.
Worked example — from (0, 0) to (3, 4)
The distance from (0, 0) to (3, 4) is d = √((3−0)² + (4−0)²) = √(9+16) = √25 = 5 — a direct 3-4-5 Pythagorean triple, with the horizontal and vertical gaps between the two points standing in for a right triangle's two legs.
Compare a pair of points with negative coordinates involved, (−2, 3) to (4, −5): d = √((4−(−2))² + (−5−3)²) = √(6² + (−8)²) = √(36+64) = √100 = 10 — the formula handles negative coordinates without any adjustment, since each difference is squared before anything else happens.
Questions
What is the Distance Formula?
d = √((x₂−x₁)² + (y₂−y₁)²), where (x₁,y₁) and (x₂,y₂) are the two points. It's a direct application of the Pythagorean theorem, treating the coordinate gaps between the two points as the two legs of a right triangle.
Does it matter which point I enter first?
No — swapping the two points flips the sign of each coordinate difference, but squaring erases the sign either way, so the resulting distance is identical regardless of which point is labeled first.
What is the distance between a point and itself?
Exactly zero — both coordinate differences vanish when the two points coincide, since there's no gap in either direction to measure.
How is this related to the Pythagorean theorem?
It IS the Pythagorean theorem, restated for coordinate points: the horizontal and vertical distances between the two points play the role of a right triangle's two legs, and the straight-line distance between them plays the role of the hypotenuse.
Can this formula be extended to three dimensions?
Yes — adding a z-coordinate difference term, (z₂−z₁)², under the same square root extends the Distance Formula to 3D space, following the identical Pythagorean logic one dimension further.