Distance Formula

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

The Formula

d=(x2βˆ’x1)2+(y2βˆ’y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

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

Quick Example

Distance between (0,0)(0,0) and (3,4)(3,4) is 32+42=5\sqrt{3^2 + 4^2} = 5

Notation

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

What This Formula Means

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

'As the crow flies'β€”the straight-line separation between two locations.

Formal View

Euclidean distance: d(P,Q)=βˆ₯Pβˆ’Qβˆ₯=βˆ‘i=1n(piβˆ’qi)2d(P,Q) = \|P - Q\| = \sqrt{\sum_{i=1}^n (p_i - q_i)^2} for P,Q∈RnP,Q \in \mathbb{R}^n; satisfies metric axioms: d(P,Q)β‰₯0d(P,Q) \geq 0, d(P,Q)=0β€…β€ŠβŸΊβ€…β€ŠP=Qd(P,Q) = 0 \iff P = Q, d(P,Q)=d(Q,P)d(P,Q) = d(Q,P), d(P,R)≀d(P,Q)+d(Q,R)d(P,R) \leq d(P,Q) + d(Q,R)

Worked Examples

Example 1

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

Answer

d=5d = 5 units

First step

1
Step 1: Use the distance formula: d=(x2βˆ’x1)2+(y2βˆ’y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.

Full solution

  1. 2
    Step 2: Substitute: d=(4βˆ’1)2+(6βˆ’2)2=32+42d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{3^2 + 4^2}.
  2. 3
    Step 3: Calculate: d=9+16=25=5d = \sqrt{9 + 16} = \sqrt{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(βˆ’2,3)P(-2, 3) and Q(4,βˆ’1)Q(4, -1). Leave your answer in simplest radical form.

Example 3

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

Common Mistakes

  • Reporting a negative distance β€” distance is always non-negative.
  • Forgetting to square the differences before adding β€” the formula is (Ξ”x)2+(Ξ”y)2\sqrt{(\Delta x)^2+(\Delta y)^2}, not Ξ”x+Ξ”y\Delta x+\Delta y.
  • Measuring along a bent path instead of straight β€” 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 β€” 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?" β€” rather than by familiar numbers β€” 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'β€”the straight-line separation between two locations.

What do the symbols mean in the Distance formula?

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

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 β€” 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?" β€” rather than by familiar numbers β€” 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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