Distance Formula Formula

Distance formula is a formula for finding the distance between two points in the coordinate plane, derived directly from the Pythagorean theorem.

The Formula

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

When to use: Imagine two points on a grid. Draw a horizontal line from one and a vertical line from the other to form a right triangle. The horizontal leg is the difference in xx-coordinates, the vertical leg is the difference in yy-coordinates, and the hypotenuse—the direct distance—comes from the Pythagorean theorem. The distance formula is just a2+b2=c2a^2 + b^2 = c^2 in coordinate clothing.

Quick Example

Distance from (1,2)(1, 2) to (4,6)(4, 6): d=(4−1)2+(6−2)2=9+16=25=5d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5

Notation

dd for distance; (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are the two points

What This Formula Means

A formula for finding the distance between two points in the coordinate plane, derived directly from the Pythagorean theorem.

Imagine two points on a grid. Draw a horizontal line from one and a vertical line from the other to form a right triangle. The horizontal leg is the difference in xx-coordinates, the vertical leg is the difference in yy-coordinates, and the hypotenuse—the direct distance—comes from the Pythagorean theorem. The distance formula is just a2+b2=c2a^2 + b^2 = c^2 in coordinate clothing.

Formal View

d(P1,P2)=(x2−x1)2+(y2−y1)2d(P_1, P_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} in R2\mathbb{R}^2; generalization: d(P,Q)=∑i=1n(qi−pi)2d(P, Q) = \sqrt{\sum_{i=1}^n (q_i - p_i)^2} in Rn\mathbb{R}^n; derived from the Pythagorean theorem

Worked Examples

Example 1

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

Answer

d=5d = 5

First step

1
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. d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.

Full solution

  1. 2
    Identify the coordinates: (x1,y1)=(1,2)(x_1, y_1) = (1, 2) and (x2,y2)=(4,6)(x_2, y_2) = (4, 6). Compute the differences: x2−x1=3x_2 - x_1 = 3, y2−y1=4y_2 - y_1 = 4.
  2. 3
    Substitute: d=32+42=9+16=25=5d = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. Recognise the 3-4-5 Pythagorean triple — no calculator needed.
The distance formula is a direct application of the Pythagorean theorem on the coordinate plane. The horizontal and vertical differences form the legs of a right triangle, and the distance is the hypotenuse.

Example 2

medium
Find the distance between (−3,5)(-3, 5) and (2,−7)(2, -7).

Example 3

medium
Show that the points A(1,1),B(4,5),C(8,2)A(1, 1), B(4, 5), C(8, 2) form a right triangle.

Common Mistakes

  • Forgetting to square the differences before adding — you must square each coordinate gap, then add, then root.
  • Adding before squaring (taking (x2−x1)+(y2−y1)\sqrt{(x_2-x_1)+(y_2-y_1)}) — square first, sum the squares, then take the root.
  • Subtracting in inconsistent order — order does not matter because the differences are squared, but mixing xx with yy does.

Why This Formula Matters

It is the Pythagorean theorem made portable across the whole coordinate plane, the tool that lets coordinate proofs verify equal sides, radii, and triangle types by computation instead of by eye. Recognizing it by "Do I have two points' coordinates and need the length of the segment joining them?" — rather than by familiar numbers — is what lets a student tell it apart from midpoint formula and slope and pythagorean theorem in a mixed problem set.

Frequently Asked Questions

What is the Distance Formula formula?

A formula for finding the distance between two points in the coordinate plane, derived directly from the Pythagorean theorem.

How do you use the Distance Formula formula?

Imagine two points on a grid. Draw a horizontal line from one and a vertical line from the other to form a right triangle. The horizontal leg is the difference in xx-coordinates, the vertical leg is the difference in yy-coordinates, and the hypotenuse—the direct distance—comes from the Pythagorean theorem. The distance formula is just a2+b2=c2a^2 + b^2 = c^2 in coordinate clothing.

What do the symbols mean in the Distance Formula formula?

dd for distance; (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) are the two points

Why is the Distance Formula formula important in Math?

It is the Pythagorean theorem made portable across the whole coordinate plane, the tool that lets coordinate proofs verify equal sides, radii, and triangle types by computation instead of by eye. Recognizing it by "Do I have two points' coordinates and need the length of the segment joining them?" — rather than by familiar numbers — is what lets a student tell it apart from midpoint formula and slope and pythagorean theorem in a mixed problem set.

What do students get wrong about Distance Formula?

The procedure for distance formula is the easy part; the trap is forgetting to square the differences before adding. Asking "Do I have two points' coordinates and need the length of the segment joining them?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Distance Formula formula?

Before studying the Distance Formula formula, you should understand: pythagorean theorem, coordinate plane, square roots.

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