What is the formula for Distance Formula?

The mathematical formula for Distance Formula is d = √((x₂-x₁)² + (y₂-y₁)²). It is used for finding the euclidean straight-line distance between two points (x₁, y₁) and (x₂, y₂).

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Distance Formula

Coordinate Geometry • Grades 8, 9, 10

Coordinate Geometry

Distance Formula

Q: What do the variables in the Distance Formula formula represent?

In the formula d = √((x₂-x₁)² + (y₂-y₁)²): d = Euclidean distance length, (x₁, y₁) = Coordinates of first point, (x₂, y₂) = Coordinates of second point.

Q: How do you calculate Distance Formula step-by-step?

To calculate distance formula: 1. Identify the given values (d = Euclidean distance length, (x₁, y₁) = Coordinates of first point, (x₂, y₂) = Coordinates of second point). 2. Substitute the values into the formula d = √((x₂-x₁)² + (y₂-y₁)²). 3. Simplify and evaluate step-by-step.

Learn how to find the exact distance between any two points on a 2D map using the Distance Formula.

The Formula

d = √(x₂-x₁)² + (y₂-y₁)²

Used for: Finding the Euclidean straight-line distance between two points (x₁, y₁) and (x₂, y₂)

dEuclidean distance length

(x₁, y₁)Coordinates of first point

(x₂, y₂)Coordinates of second point

⚡Live Interactive Quick Solver

Instant Calculation

Point 1 (x₁, y₁):

Point 2 (x₂, y₂):

Distance (d):5.00 units

Step: d = √((4 - 1)² + (6 - 2)²) = √(25) = 5.00

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

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🎯 Key Concepts

Understand the geometry behind the distance:

📐Pythagorean Origin

The formula is actually the Pythagorean theorem applied to the coordinate plane.

📍Coordinate Plane

Calculates the straight-line distance between any two points (x₁, y₁) and (x₂, y₂).

📏Always Positive

Because the differences are squared, the result is always a non-negative distance.

📝 Worked Examples

Example 1: Positive Coordinates

Problem: Find the distance between P₁(1, 2) and P₂(4, 6).

Identify Pointsx₁=1, y₁=2 and x₂=4, y₂=6

Substitute valuesd = √((4 - 1)² + (6 - 2)²)

Subtractd = √(3² + 4²)

Square and Addd = √(9 + 16) = √25

Answer: d = 5 units

Example 2: Negative Coordinates

Problem: Find the distance between (-1, -1) and (2, 3).

Identify Pointsx₁=-1, y₁=-1 and x₂=2, y₂=3

Substitute valuesd = √((2 - (-1))² + (3 - (-1))²)

Simplifyd = √(3² + 4²)

Calculated = √(9 + 16) = √25 = 5

Answer: d = 5 units

💡 Pro Tips

Point Order

It doesn't matter which point you call (x₁,y₁) and which you call (x₂,y₂). Squaring makes the result the same.

Vertical/Horizontal

If the x-coordinates are the same, the distance is just the difference in y-coordinates (and vice versa).

Visualization

Draw a right triangle between the points to understand why the formula works.

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