Distance Calculator
Calculate the distance between two points using the distance formula: d = √[(x₂-x₁)² + (y₂-y₁)²]
Enter Two Points to Find the Distance
Step-by-Step Solution:
1. Using the distance formula: d = √[(x₂-x₁)² + (y₂-y₁)²]
2. Substitute values: d = √[(3-0)² + (4-0)²]
3. Calculate differences: d = √[3² + 4²]
4. Square the differences: d = √[9 + 16]
5. Add: d = √25
6. Take square root: d = 5
What is the Distance Formula?
The distance formula finds the straight-line distance (Euclidean distance) between any two points in a coordinate plane.
Distance Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
where (x₁,y₁) is the first point and (x₂,y₂) is the second point.
How to Find Distance Between Two Points
- Identify coordinates: Label your points as (x₁,y₁) and (x₂,y₂)
- Find horizontal distance: Calculate (x₂ - x₁)
- Find vertical distance: Calculate (y₂ - y₁)
- Square both differences: (x₂ - x₁)² and (y₂ - y₁)²
- Add the squares: (x₂ - x₁)² + (y₂ - y₁)²
- Take the square root: √[(x₂ - x₁)² + (y₂ - y₁)²]
Why the Distance Formula Works
The distance formula comes from the Pythagorean theorem (a² + b² = c²). When you plot two points and draw a right triangle: - One leg is the horizontal distance (x₂ - x₁) - The other leg is the vertical distance (y₂ - y₁) - The hypotenuse is the distance between the points
Real-World Applications
The distance formula is used in: - GPS and Navigation: Calculating routes and distances - Computer Graphics: Rendering and collision detection - Physics: Motion and trajectory calculations - Engineering: Design and measurement - Game Development: Character movement and AI
Common Questions About Distance
What is the distance formula?
The distance formula calculates the straight-line distance between two points: d = √[(x₂-x₁)² + (y₂-y₁)²]. It's derived from the Pythagorean theorem.
When do you use the distance formula?
Use the distance formula in coordinate geometry to find the length of a line segment between two points, in navigation, physics, computer graphics, and real-world distance problems.
Is distance always positive?
Yes, distance is always positive or zero. The square root in the formula ensures this, as we're measuring the length between points.