Distance Formula Guide: 2D, 3D & Coordinate Planes

Distance Formula — Direct Answer

The distance formula is d = √((x₂−x₁)² + (y₂−y₁)²). It finds the straight-line distance between two points on a coordinate plane. For points (3, 4) and (7, 8): d = √(16 + 16) = 5.66 units. The formula derives from the Pythagorean theorem.

→ Calculate distance between points with our free calculator

How to Use the Distance Formula

The formula is a rearranged Pythagorean theorem applied to coordinate deltas.

2D: d = √((x₂−x₁)² + (y₂−y₁)²)
3D: d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²)

Where:

• (x₁, y₁) = first point
• (x₂, y₂) = second point
• d = distance between them

How to identify the coordinates

Every coordinate pair follows the format (x, y). The x is horizontal, y is vertical.

Example: Point A = (2, 3), Point B = (8, 11)

How to calculate delta x and delta y

Δx = x₂ − x₁
Δy = y₂ − y₁

Example: (2, 3) to (8, 11):

• Δx = 8 − 2 = 6
• Δy = 11 − 3 = 8

How to apply the formula

Example: (2, 3) to (8, 11)

• Δx = 6, Δx² = 36
• Δy = 8, Δy² = 64
• Sum = 100
• d = √100 = 10 units

Example: (0, 0) to (3, 4)

• Δx = 3, Δx² = 9
• Δy = 4, Δy² = 16
• Sum = 25
• d = √25 = 5 units (classic 3-4-5 triangle)

Distance Formula Chart by Point Coordinates

Point 1	Point 2	Δx	Δy	Distance
(0, 0)	(3, 4)	3	4	5.00
(0, 0)	(6, 8)	6	8	10.00
(0, 0)	(5, 12)	5	12	13.00
(1, 2)	(4, 6)	3	4	5.00
(2, 3)	(8, 11)	6	8	10.00
(1, 1)	(10, 10)	9	9	12.73
(−2, 3)	(4, −5)	6	−8	10.00
(0, 0)	(1, 1)	1	1	1.41

3D distance examples

Point 1	Point 2	Distance
(0, 0, 0)	(2, 3, 6)	7.00
(1, 2, 3)	(4, 6, 3)	5.00
(0, 0, 0)	(1, 1, 1)	1.73

Distance Formula Applications

Construction layout

Distance between two stakes on a plot:

• Stake A at (0, 0)
• Stake B at (30, 40)
• Distance = √(900 + 1,600) = 50 ft

GPS coordinates (approximate)

For small distances, convert latitude/longitude to feet and use the formula. For large distances, use the Haversine formula.

Video game graphics

Every 3D game engine uses the distance formula to detect collisions.

Physics vectors

Force magnitude, velocity, and displacement all use this formula.

Common Distance Formula Mistakes

Mistake 1: Wrong sign on negative coordinates

(−3)² = 9, not −9. Squaring always produces positive values.

Fix: Square the delta, not the raw coordinate. Always positive.

Mistake 2: Forgetting the square root

Many people calculate Δx² + Δy² and stop. The answer is the square root.

Fix: Final step is √. Without it, you have d², not d.

Mistake 3: Using the slope formula instead

Slope is (y₂−y₁)/(x₂−x₁) — a ratio, not a distance.

Fix: Slope gives steepness. Distance formula gives length.

Distance vs Midpoint vs Slope

Distance formula

• Formula: d = √((x₂−x₁)² + (y₂−y₁)²)
• Returns: Length between points
• Unit: Same as coordinates

Midpoint formula

• Formula: ((x₁+x₂)/2, (y₁+y₂)/2)
• Returns: Center point
• Use: Finding middle

Slope formula

• Formula: m = (y₂−y₁)/(x₂−x₁)
• Returns: Steepness
• Use: Line direction

Distance Formula FAQ

What is the distance formula?

d = √((x₂−x₁)² + (y₂−y₁)²). It calculates the straight-line distance between two points on a coordinate plane.

How do you find the distance between two points?

Subtract the x-values, square; subtract the y-values, square; add both; take the square root.

What is the distance between (0, 0) and (3, 4)?

5 units. √(9 + 16) = √25 = 5.

Is the distance formula the same as the Pythagorean theorem?

Yes. The distance formula is the Pythagorean theorem applied to coordinate deltas.

What is the 3D distance formula?

d = √((x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²). Add the z-axis delta squared to the 2D formula.

Next Steps for Coordinate Math

• Distance Formula Calculator — Any two points
• Triangle Calculator — Triangle sides
• Math Calculator — All math tools
• Pythagorean Theorem Guide — Right triangles
• Slope Formula Guide — Line math

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Last updated October 2026. Works in any unit — just keep coordinates consistent.