Distance Formula Examples with Answers- Practice Problems

What Is the Distance Formula?

The distance formula finds the length between two points on a coordinate plane. It's derived from the Pythagorean theorem and works every time, no exceptions.

Here's the formula:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

That's it. Two points go in, distance comes out. The squaring eliminates negative numbers, so you don't have to worry about which point is "first."

How the Distance Formula Works

You're calculating the hypotenuse of a right triangle. The difference in x-values is one leg, the difference in y-values is the other. The distance formula is just the Pythagorean theorem packaged for coordinates.

Steps:

• Subtract the x-coordinates, then square the result
• Subtract the y-coordinates, then square the result
• Add those two squares together
• Take the square root of that sum

Distance Formula Examples with Solutions

Example 1: Basic Points

Find the distance between (3, 4) and (7, 1).

Apply the formula:

d = √[(7 - 3)² + (1 - 4)²]

d = √[(4)² + (-3)²]

d = √[16 + 9]

d = √25 = 5

Answer: 5 units

Example 2: Points with Negative Coordinates

Find the distance between (-2, -5) and (4, 3).

d = √[(4 - (-2))² + (3 - (-5))²]

d = √[(6)² + (8)²]

d = √[36 + 64]

d = √100 = 10

Answer: 10 units

Example 3: Origin to a Point

Find the distance from (0, 0) to (5, 12).

When one point is the origin, the formula simplifies:

d = √[(5 - 0)² + (12 - 0)²]

d = √[25 + 144]

d = √169 = 13

Answer: 13 units

Example 4: Points with Same X-Coordinate

Find the distance between (2, 1) and (2, 7).

When x-coordinates match, you're just measuring vertical distance:

d = √[(2 - 2)² + (7 - 1)²]

d = √[0 + 36]

d = √36 = 6

Answer: 6 units

Practice Problems

Try these on your own before checking the answers.

Problem 1: Find the distance between (1, 2) and (4, 6).

Problem 2: Find the distance between (-3, 4) and (2, -1).

Problem 3: Find the distance between (0, -4) and (3, 0).

Problem 4: Find the distance between (-5, -2) and (-1, -2).

Answers

Problem 1: d = √[(4-1)² + (6-2)²] = √[9 + 16] = √25 = 5

Problem 2: d = √[(2-(-3))² + (-1-4)²] = √[25 + 25] = √50 ≈ 7.07

Problem 3: d = √[(3-0)² + (0-(-4))²] = √[9 + 16] = √25 = 5

Problem 4: d = √[(-1-(-5))² + (-2-(-2))²] = √[16 + 0] = 4

Distance Formula vs. Other Methods

Common Mistakes to Avoid

• Forgetting to square the differences — the squares are non-negotiable
• Taking the square root too early — add the squares first, then root
• Sign errors when subtracting — double-check your negatives
• Rounding too early — keep full precision until the final answer

How to Get Fast at the Distance Formula

Most students waste time rewriting the formula for each problem. Don't.

Step 1: Identify (x₁, y₁) and (x₂, y₂) — label them immediately

Step 2: Calculate x₂ - x₁ and y₂ - y₁ in your head

Step 3: Square both differences

Step 4: Add the squares

Step 5: Take the square root

Practice 10 problems back-to-back and you'll have it memorized. The formula doesn't change, and neither does the process.

When You'll Actually Use This

Geometry class is one thing. But the distance formula shows up in:

• Navigation apps calculating travel distances
• Computer graphics and game development
• Data analysis clustering nearby points
• Physics problems involving displacement

It's not just academic busywork. Programmers use it constantly for collision detection, proximity alerts, and spatial queries.

Quick Reference

Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Key fact: The result is always positive. Distance has no direction.

Special cases:

• Horizontal line: d = |x₂ - x₁|
• Vertical line: d = |y₂ - y₁|
• Through origin: d = √(x² + y²)