Distance Formula in Mathematics- Examples and Practice
What Is the Distance Formula?
The distance formula finds the straight-line distance between two points on a coordinate plane. It comes straight from the Pythagorean theorem, so if you know that, you already understand half of this.
You get the difference between x-coordinates, square it. Get the difference between y-coordinates, square that. Add them together, then take the square root. That's it.
The Formula
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
The subscripts just tell you which point is first and which is second. It doesn't matter which point you call 1 and which you call 2—the distance stays the same.
How to Use It (Step by Step)
Let's find the distance between points (2, 3) and (8, 11).
Step 1: Subtract the x-coordinates: 8 - 2 = 6
Step 2: Subtract the y-coordinates: 11 - 3 = 8
Step 3: Square both differences: 6² = 36, 8² = 64
Step 4: Add them: 36 + 64 = 100
Step 5: Take the square root: √100 = 10
The distance is 10 units.
Common Mistakes to Avoid
- Forgetting to square the differences before adding
- Taking the square root at the wrong time
- Mixing up which coordinate goes with which point
- Not using absolute values when points have negative coordinates (though the formula handles this anyway)
Examples With Negative Numbers
Find the distance between (-4, 2) and (3, -6).
Step 1: 3 - (-4) = 7
Step 2: -6 - 2 = -8
Step 3: 7² = 49, (-8)² = 64
Step 4: 49 + 64 = 113
Step 5: √113 ≈ 10.63
The negatives cancel out when you square. You don't need to worry about them.
Distance Formula vs. Other Methods
| Method | Best For | Formula |
|---|---|---|
| Distance Formula | Any two points on a coordinate plane | √[(x₂-x₁)² + (y₂-y₁)²] |
| Pythagorean Theorem | Right triangles, geometric proofs | a² + b² = c² |
| Slope Formula | Finding steepness, not distance | (y₂-y₁)/(x₂-x₁) |
| Midpoint Formula | Finding the center point between two locations | ((x₁+x₂)/2, (y₁+y₂)/2) |
3D Distance Formula
When you have points in three dimensions, add the z-coordinate difference:
d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
Example: Find distance between (1, 2, 3) and (4, 6, 7).
3² + 4² + 4² = 9 + 16 + 16 = 41
√41 ≈ 6.4
The same process, just one more step.
Practice Problems
Try these before checking the answers below:
- Find distance between (0, 0) and (5, 12)
- Find distance between (-2, -3) and (4, 1)
- Find distance between (1, 1) and (1, 6)
Answers:
- 13 units
- √52 ≈ 7.21 units
- 5 units (straight vertical line)
Where This Actually Shows Up
- GPS and mapping — calculating actual distances between coordinates
- Video game collision detection — checking if objects are within range
- Architecture — measuring diagonal distances in building design
- Physics — displacement calculations
Quick Reference
When you need to find distance between two points:
- Subtract x's, square it
- Subtract y's, square it
- Add the squares
- Square root the sum
That's the entire formula. Memorize the pattern, not the letters. Once you see it's just Pythagorean theorem in disguise, it clicks.