Distance Formula

The Formula

d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^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 \sqrt{3^2 + 4^2} = 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\| = \sqrt{\sum_{i=1}^n (p_i - q_i)^2} for P,Q \in \mathbb{R}^n; satisfies metric axioms: d(P,Q) \geq 0, d(P,Q) = 0 \iff P = Q, d(P,Q) = d(Q,P), d(P,R) \leq d(P,Q) + d(Q,R)

Worked Examples

Example 1

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

Solution

  1. 1
    Step 1: Use the distance formula: d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.
  2. 2
    Step 2: Substitute: d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{3^2 + 4^2}.
  3. 3
    Step 3: Calculate: d = \sqrt{9 + 16} = \sqrt{25} = 5.

Answer

d = 5 units
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.

Common Mistakes

  • Computing (x_2 - x_1) + (y_2 - y_1) instead of using the Pythagorean theorem — that gives the taxicab distance, not straight-line distance
  • Forgetting to take the square root after computing (\Delta x)^2 + (\Delta y)^2
  • Reporting a negative distance — distance is always non-negative

Why This Formula Matters

Fundamental measurement in geometry, physics, and navigation.

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?

Fundamental measurement in geometry, physics, and navigation.

What do students get wrong about Distance?

Distance on a plane uses the Pythagorean theorem (distance formula).

What should I learn before the Distance formula?

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

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