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Distance Formula
Coordinate Geometry • Grades 8, 9, 10
Coordinate Geometry
Distance Formula
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 CalculationDistance (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 OriginThe formula is actually the Pythagorean theorem applied to the coordinate plane.
📍
Coordinate PlaneCalculates the straight-line distance between any two points (x₁, y₁) and (x₂, y₂).
📏
Always PositiveBecause 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).1
Identify Points
x₁=1, y₁=2 and x₂=4, y₂=62
Substitute values
d = √((4 - 1)² + (6 - 2)²)3
Subtract
d = √(3² + 4²)4
Square and Add
d = √(9 + 16) = √25Answer: d = 5 units
Example 2: Negative Coordinates
Problem:
Find the distance between (-1, -1) and (2, 3).1
Identify Points
x₁=-1, y₁=-1 and x₂=2, y₂=32
Substitute values
d = √((2 - (-1))² + (3 - (-1))²)3
Simplify
d = √(3² + 4²)4
Calculate
d = √(9 + 16) = √25 = 5Answer: 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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