Distance Formula

Distance is the length of the shortest path between two points, always a non-negative real number.

The Formula

d=(x2−x1)2+(y2−y1)2

When to use: 'As the crow flies'—the straight-line separation between two locations.

Quick Example

Distance between (0,0) and (3,4) is 32+42=5

Notation

d(A,B) or ∣AB∣ denotes the distance between points A and B

What This Formula Means

The length of the shortest path between two points, always a non-negative real number.

'As the crow flies'—the straight-line separation between two locations.

Formal View

Euclidean distance: d(P,Q)=∥P−Q∥=∑i=1n(pi−qi)2 for P,Q∈Rn; satisfies metric axioms: d(P,Q)≥0, d(P,Q)=0  ⟺  P=Q, d(P,Q)=d(Q,P), d(P,R)≤d(P,Q)+d(Q,R)

Worked Examples

Example 1

easy
Find the distance between points A(1,2) and B(4,6).

Answer

d=5 units

First step

1
Step 1: Use the distance formula: d=(x2−x1)2+(y2−y1)2.

Full solution

  1. 2
    Step 2: Substitute: d=(4−1)2+(6−2)2=32+42.
  2. 3
    Step 3: Calculate: d=9+16=25=5.
The distance formula is derived from the Pythagorean theorem. The horizontal and vertical separations form the legs of a right triangle, and the distance is the hypotenuse. Here the 3-4-5 right triangle makes the answer a whole number.

Example 2

medium
Find the distance between P(−2,3) and Q(4,−1). Leave your answer in simplest radical form.

Example 3

medium
Show that the points (0,0),(5,0),(5,5) form a right isosceles triangle. Compute all side lengths.

Common Mistakes

  • Reporting a negative distance — distance is always non-negative.
  • Forgetting to square the differences before adding — the formula is (Δx)2+(Δy)2, not Δx+Δy.
  • Measuring along a bent path instead of straight — distance is the shortest, straight-line length.

Why This Formula Matters

Distance turns the Pythagorean theorem into a coordinate tool and underpins the distance formula, circles (fixed distance from a center), and all later geometry that measures separation — it is how 'how far apart' becomes a precise number. Recognizing it by "Am I finding the shortest straight-line length between two specific points?" — rather than by familiar numbers — is what lets a student tell it apart from displacement and perimeter and distance along a path in a mixed problem set.

Frequently Asked Questions

What is the Distance formula?

The length of the shortest path between two points, always a non-negative real number.

How do you use the Distance formula?

'As the crow flies'—the straight-line separation between two locations.

What do the symbols mean in the Distance formula?

d(A,B) or ∣AB∣ denotes the distance between points A and B

Why is the Distance formula important in Math?

Distance turns the Pythagorean theorem into a coordinate tool and underpins the distance formula, circles (fixed distance from a center), and all later geometry that measures separation — it is how 'how far apart' becomes a precise number. Recognizing it by "Am I finding the shortest straight-line length between two specific points?" — rather than by familiar numbers — is what lets a student tell it apart from displacement and perimeter and distance along a path in a mixed problem set.

What do students get wrong about Distance?

The procedure for distance is the easy part; the trap is reporting a negative distance. Asking "Am I finding the shortest straight-line length between two specific points?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Distance formula?

Before studying the Distance formula, you should understand: pythagorean theorem.

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