How to Find Distance of Two Points? (+FREE Worksheet!)

Find the distance between two points on a coordinate plane using their coordinates and the distance formula.

The distance formula lets you calculate the exact length of any line segment on a coordinate plane using only the coordinates of its two endpoints. It comes directly from the Pythagorean theorem, so understanding one makes the other intuitive. Mastering this formula is a key skill for coordinate geometry throughout Algebra 1 and geometry.

What Is the Distance Formula?

For two points \(\color{blue}{(x_{1}, y_{1})}\) and \(\color{blue}{(x_{2}, y_{2})}\), the distance \(\color{blue}{d}\) between them is:

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d = √((x₂ − x₁)² + (y₂ − y₁)²)

You are finding the hypotenuse of a right triangle whose legs are the horizontal and vertical distances between the two points.

How to Use the Distance Formula

Step 1: Label the coordinates

Identify \(\color{blue}{(x_{1}, y_{1})}\) and \(\color{blue}{(x_{2}, y_{2})}\).

Step 2: Subtract x-coordinates and y-coordinates

Compute \(\color{blue}{x_{2} – x_{1}}\) and \(\color{blue}{y_{2} – y_{1}}\).

Step 3: Square each difference

\(\color{blue}{(x_{2} – x_{1})^{2}}\) and \(\color{blue}{(y_{2} – y_{1})^{2}}\). Note: squaring removes negative signs.

Step 4: Add and take the square root

Add the two squared differences and take the square root. Simplify if the result is a perfect square.

• Distance from \(\color{blue}{(0, 0)}\) to \(\color{blue}{(3, 4)}\): \(\color{blue}{d = \sqrt{(9 + 16)} = \sqrt{25} = 5}\)

Step-by-Step Summary

1. Write down both endpoints and label them.
2. Find \(\color{blue}{(x_{2} – x_{1})}\) and \(\color{blue}{(y_{2} – y_{1})}\).
3. Square both differences.
4. Add the squares.
5. Take the positive square root of the sum.
6. Simplify if possible.

Watch: Distance Formula (Video Lesson)

Khan Academy derives the distance formula from the Pythagorean theorem and works through examples:

Finding Distance of Two Points – Worked Examples

Example 1: Find the distance between \(\color{blue}{(0, 0)}\) and \(\color{blue}{(3, 4)}\).

\(\color{blue}{d = \sqrt{((3 – 0)}^{2} + (4 – 0)^{2}) = \sqrt{(9 + 16)} = \sqrt{25} = 5}\)
Answer: \(\color{blue}{5}\)

Example 2: Find the distance between \(\color{blue}{(1, 2)}\) and \(\color{blue}{(4, 6)}\).

\(\color{blue}{d = \sqrt{((4 – 1)}^{2} + (6 – 2)^{2}) = \sqrt{(9 + 16)} = \sqrt{25} = 5}\)
Answer: \(\color{blue}{5}\)

Example 3: Find the distance between \(\color{blue}{(-1, 3)}\) and \(\color{blue}{(2, 7)}\).

\(\color{blue}{d = \sqrt{((2 – (-1)})^{2} + (7 – 3)^{2}) = \sqrt{(9 + 16)} = \sqrt{25} = 5}\)
Answer: \(\color{blue}{5}\)

Example 4: Find the distance between \(\color{blue}{(-3, -4)}\) and \(\color{blue}{(3, 4)}\).

\(\color{blue}{d = \sqrt{((3 – (-3)})^{2} + (4 – (-4))^{2}) = \sqrt{(36 + 64)} = \sqrt{100} = 10}\)
Answer: \(\color{blue}{10}\)

More Practice: Distance Formula Video

This video works through finding the distance between two coordinate points with full step-by-step solutions:

Exercises for Finding the Distance of Two Points

Find the distance between each pair of points. Simplify all square roots.

1. \(\color{blue}{(0, 0)}\) and \(\color{blue}{(5, 12)}\)
2. \(\color{blue}{(2, 3)}\) and \(\color{blue}{(5, 7)}\)
3. \(\color{blue}{(-1, -1)}\) and \(\color{blue}{(2, 3)}\)
4. \(\color{blue}{(0, 4)}\) and \(\color{blue}{(3, 0)}\)
5. \(\color{blue}{(-2, 1)}\) and \(\color{blue}{(2, 4)}\)
6. \(\color{blue}{(1, 1)}\) and \(\color{blue}{(7, 9)}\)

Answers

1. \(\color{blue}{13}\)
2. \(\color{blue}{5}\)
3. \(\color{blue}{5}\)
4. \(\color{blue}{5}\)
5. \(\color{blue}{5}\)
6. \(\color{blue}{10}\)

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Frequently Asked Questions

What is the formula for finding distance between two points?

The distance formula is d = √((x₂ − x₁)² + (y₂ − y₁)²). Subtract the x-coordinates, subtract the y-coordinates, square both differences, add them and take the square root. For the points (1, 2) and (4, 6), the differences are 3 and 4, so d = √(9 + 16) = √25 = 5 units.

How do you find the distance between two points on a coordinate plane?

Label one point (x₁, y₁) and the other (x₂, y₂), then plug them into the distance formula. For (−2, 5) and (4, −3): x₂ − x₁ = 4 − (−2) = 6 and y₂ − y₁ = −3 − 5 = −8. Squaring gives 36 and 64, which add to 100, and √100 = 10. It does not matter which point you call first, because squaring removes any negative sign.

How do you find the distance between two points without the distance formula?

Use the Pythagorean theorem directly, which is where the distance formula comes from. Plot both points, draw a right triangle with one horizontal and one vertical leg, and count the length of each leg. Then use a² + b² = c². From (0, 0) to (3, 4), the legs are 3 and 4, so c² = 9 + 16 = 25 and the distance is 5.

How do you find the distance between points on a horizontal or vertical line?

When two points share the same y-coordinate, the line is horizontal, so the distance is the absolute value of the difference in x-coordinates. When they share an x-coordinate, subtract the y-coordinates instead. From (2, 7) to (9, 7), the distance is |9 − 2| = 7. From (−3, −1) to (−3, 6), it is |6 − (−1)| = 7. No square roots are needed.

How do you find the distance between points in 3D?

In three dimensions, add a third term for the z-coordinates: d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²). For (1, 2, 3) and (3, 5, 9), the differences are 2, 3 and 6, so d = √(4 + 9 + 36) = √49 = 7. It is the same idea as the 2D formula, just applied along one more axis.

What is the distance between the points (0, 0) and (36, 15)?

The distance is 39 units. Using the distance formula, d = √((36 − 0)² + (15 − 0)²) = √(1,296 + 225) = √1,521 = 39. When one point is the origin, the formula shortens to √(x² + y²). These numbers form a 5-12-13 right triangle scaled up by 3, since 15 = 3 × 5, 36 = 3 × 12 and 39 = 3 × 13.

Related Topics

• Finding Midpoint
• One-Step Equations
• Multi-Step Equations
• Evaluating Two Variables
• Slope of a Line