📌 Quick Answer
The distance formula gives the straight-line distance between two points (x₁, y₁) and (x₂, y₂) in a plane: d = √[(x₂ − x₁)² + (y₂ − y₁)²].
It is derived directly from the Pythagorean theorem.
🔹 Key Takeaways
- d = √[(x₂ − x₁)² + (y₂ − y₁)²] in 2D.
- It comes straight from the Pythagoras theorem.
- 3D version: d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²].
- Distance is always positive.
What Is the Distance Formula?
The distance formula calculates the straight-line distance between two points in a coordinate plane. For points A(x₁, y₁) and B(x₂, y₂):
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Derivation from the Pythagorean Theorem
Plot the two points and form a right-angled triangle with horizontal side (x₂ − x₁) and vertical side (y₂ − y₁). The distance d is the hypotenuse, so by Pythagoras d² = (x₂ − x₁)² + (y₂ − y₁)², giving the distance formula on taking the square root.
Distance Formula in 3D
For two points in space, A(x₁, y₁, z₁) and B(x₂, y₂, z₂):
d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²]
Worked Example
Find the distance between (1, 2) and (4, 6): d = √[(4−1)² + (6−2)²] = √[9 + 16] = √25 = 5 units.
Frequently Asked Questions
What is the distance formula?
It gives the distance between two points (x₁,y₁) and (x₂,y₂): d = √[(x₂−x₁)² + (y₂−y₁)²].
How is the distance formula derived?
From the Pythagorean theorem: the horizontal and vertical gaps between the points form the legs of a right triangle and the distance is the hypotenuse.
What is the distance formula in 3D?
d = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²] for points in three-dimensional space.
Can distance be negative?
No. Distance is always positive because it is the square root of a sum of squares.
1 thought on “Distance Formula – Details, Definition, & Derivation”