Distance (d) =
d = √(("+x2+"-"+x1+")2 + ("+y2+"-"+y1+")2)
d = √(("+dx+")2 + ("+dy+")2)
d = √("+ (dx*dx).toFixed(4) +" + "+ (dy*dy).toFixed(4) +")
d = √("+squaredSum.toFixed(4)+")
d ≈ "+distance.toFixed(6)+"";}else{squaredSum=(dx*dx)+(dy*dy)+(dz*dz);distance=Math.sqrt(squaredSum);stepHtml="d = √((x2-x1)2 + (y2-y1)2 + (z2-z1)2)
d = √(("+x2+"-"+x1+")2 + ("+y2+"-"+y1+")2 + ("+z2+"-"+z1+")2)
d = √(("+dx+")2 + ("+dy+")2 + ("+dz+")2)
d = √("+ (dx*dx).toFixed(4) +" + "+ (dy*dy).toFixed(4) +" + "+ (dz*dz).toFixed(4) +")
d = √("+squaredSum.toFixed(4)+")
d ≈ "+distance.toFixed(6)+"";}document.getElementById('resultValue').innerHTML=distance.toFixed(6);if(showSteps){document.getElementById('solutionSteps').innerHTML=stepHtml;document.getElementById('solutionSteps').style.display='block';}else{document.getElementById('solutionSteps').style.display='none';}}
Calculator Use
The distance formula calculator is a specialized tool designed to compute the straight-line distance between two points in a 2D or 3D coordinate system. Whether you are working on a geometry homework assignment or calculating physical paths in engineering, this tool provides instant and accurate results.
To use this calculator, simply choose your dimension (2D or 3D) and enter the coordinates for Point 1 and Point 2. Click "Calculate Distance" to see the numeric result and optional step-by-step breakdown.
- x1, y1, z1
- These represent the horizontal (x), vertical (y), and depth (z) coordinates of your starting position (Point 1).
- x2, y2, z2
- These represent the coordinates of your ending position (Point 2).
- Show Solution Steps
- Checking this box reveals the mathematical operations performed by the distance formula calculator, which is helpful for learning the process.
How It Works
The distance formula is essentially the Pythagorean Theorem applied to a coordinate plane. By finding the difference between the x-coordinates and y-coordinates, we create two sides of a right triangle. The distance between the points is the hypotenuse.
The fundamental formula for 2D distance used by this calculator is:
d = √[(x² – x¹)² + (y² – y¹)²]
For three-dimensional space, the formula expands to include the z-axis:
d = √[(x² – x¹)² + (y² – y¹)² + (z² – z¹)²]
- Subtraction: We subtract coordinates to find the "delta" or difference (horizontal and vertical gaps).
- Squaring: We square the differences to ensure they are positive and to satisfy the theorem.
- Summation: We add the squared values together.
- Square Root: We take the square root of the sum to find the final distance.
Calculation Example
Example: Find the distance between Point A (2, 3) and Point B (5, 7) using the 2D distance formula calculator logic.
Step-by-step solution:
- Identify coordinates: (x1=2, y1=3) and (x2=5, y2=7)
- Subtract x-values: (5 – 2) = 3
- Subtract y-values: (7 – 3) = 4
- Square the results: 3² = 9 and 4² = 16
- Add them: 9 + 16 = 25
- Take the square root: √25 = 5
- Result = 5 units
Common Questions
Can the distance be a negative number?
No. Because the differences between coordinates are squared, the result inside the square root is always zero or positive. Geometrically, distance represents a physical length, which cannot be less than zero.
What units does the distance formula calculator use?
This calculator provides a unitless numeric result. The units depend entirely on your input. If your coordinates are in meters, the distance is in meters; if they are in feet, the result is in feet.
Does the order of the points matter?
No. If you swap Point 1 and Point 2, the sign of the differences will change (e.g., from 3 to -3), but since we square these values, the outcome remains identical. The distance from A to B is always the same as the distance from B to A.