Distance Formula
Calculate distance between points
Distance Formula: d = √[(x₂-x₁)² + (y₂-y₁)²]
Calculate distance between two points
Distance Result
About Distance Formula Calculator
The Distance Formula Calculator helps you find the straight-line distance between any two points in a 2D coordinate plane. Whether you're a student learning geometry, a teacher preparing lessons, or a professional in engineering, this tool makes distance calculations instant and accurate.
Beyond just distance, our calculator also provides the midpoint, slope, and changes in x and y coordinates — everything you need for complete coordinate geometry analysis.
How to Use This Distance Formula Calculator
Step 1: Enter the x₁ and y₁ coordinates of the first point.
Step 2: Enter the x₂ and y₂ coordinates of the second point.
Step 3: Click "Calculate Distance" to see the results.
Step 4: View distance, midpoint, slope, and coordinate changes.
Step 5: Use the Reset button to clear all coordinates and start a new calculation.
Tip: Use the Quick Examples buttons to try pre-filled coordinates.
Real-World Applications of Distance Formula
📍 Navigation & GPS
GPS systems calculate straight-line distance between coordinates to determine shortest routes and estimate travel times.
📐 Construction & Surveying
Surveyors use distance formula to measure land boundaries, building layouts, and property lines accurately.
🎮 Game Development
Game engines use distance formula for collision detection, enemy AI range, and camera positioning.
✈️ Aviation & Maritime
Pilots and ship captains use great-circle distance (based on spherical coordinates) for route planning.
3D Distance Formula (Extension to Three Dimensions)
d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]
For 3D points (x₁,y₁,z₁) and (x₂,y₂,z₂). Add the squared difference in z-coordinates to the 2D formula. Used in 3D modeling, physics simulations, and engineering design.
Advanced Coordinate Geometry Concepts
Midpoint Formula
M = ((x₁+x₂)/2, (y₁+y₂)/2). Finds the point exactly halfway between two points. Useful for line bisectors and finding centers.
Slope Formula
m = (y₂-y₁)/(x₂-x₁). Measures steepness of a line. Positive slope = upward, negative = downward, zero = horizontal, undefined = vertical.
Distance Formula Proof (Derived from Pythagorean Theorem)
The distance formula is derived from the Pythagorean theorem: a² + b² = c².
In coordinate geometry, the horizontal leg is Δx = |x₂-x₁|, vertical leg is Δy = |y₂-y₁|, and the straight-line distance is the hypotenuse. Therefore, d² = (Δx)² + (Δy)², so d = √[(Δx)² + (Δy)²].
Step-by-Step Example
Find distance between (1,2) and (4,6):
1. Δx = 4 - 1 = 3
2. Δy = 6 - 2 = 4
3. Δx² = 9, Δy² = 16
4. Sum = 9 + 16 = 25
5. d = √25 = 5 units
Midpoint = (2.5, 4). Slope = 4/3 ≈ 1.33
Distance Formula
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Also finds: Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope = (y₂ - y₁) / (x₂ - x₁)
Quick Examples
Frequently Asked Questions
The distance formula calculates the straight-line distance between two points: d = √[(x₂ - x₁)² + (y₂ - y₁)²]. Derived from the Pythagorean theorem.
Subtract x coordinates (Δx), subtract y coordinates (Δy), square both, add them, then take square root. Example: (1,2) and (4,6) → Δx=3, Δy=4 → d = √(9+16) = √25 = 5.
Distance is the total path length (scalar). Displacement is straight-line distance from start to end (vector). The distance formula gives displacement, not total path length.
No, distance is always positive because we square the differences before adding them. The formula uses absolute differences.
For 3D coordinates (x₁,y₁,z₁) and (x₂,y₂,z₂): d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]. Add the squared difference in z-coordinates. Used in 3D modeling, physics, and engineering.
1) Locate both points on coordinate plane. 2) Count horizontal distance (Δx) and vertical distance (Δy). 3) Apply distance formula: √(Δx² + Δy²). On graph paper, you can also draw right triangle and measure hypotenuse.
Formula: d = |Ax₁ + By₁ + C| ÷ √(A² + B²) for line Ax + By + C = 0. Example: Distance from (3,4) to line 3x+4y-5=0 = |9+16-5| ÷ 5 = 20÷5 = 4 units.
Use Haversine formula: d = 2R × arcsin(√[sin²(Δφ/2) + cos φ₁ cos φ₂ sin²(Δλ/2)]). R = Earth's radius (6,371 km). This gives great-circle distance between two points on Earth's surface.
Euclidean distance = straight line (our formula). Manhattan distance = |x₂-x₁| + |y₂-y₁| (city block distance). Example: (0,0) to (3,4): Euclidean=5, Manhattan=7. Manhattan used in grid-based pathfinding.
For perfect squares: Check if Δx² + Δy² is perfect square (0,1,4,9,16,25,36,49,64,81,100). For (3,4): 9+16=25, √25=5. For non-perfect squares, leave as √(sum). Example: (1,2) to (4,6) gives √(9+16)=√25=5.