The first point supplies x₁ and y₁. The second point supplies x₂ and y₂.
Free Interactive Coordinate Geometry Tool
Distance Formula Practice
Learn how to find the distance between two points by labeling the coordinates, substituting into the distance formula, simplifying each difference, and finding the exact distance step by step.
Start practicingNeed Help Understanding the Distance Formula?
Learn where the distance formula comes from, how to identify x₁, y₁, x₂, and y₂, how to substitute negative coordinates correctly, and how the formula connects to the Pythagorean theorem.
Read the Distance Formula Guide
Practice Mode
Work through the entire distance formula from
substitution to the final answer.
Problem 1 of 10
10%
Current Problem
Use the labeled coordinates to substitute into the distance formula.
Full Solution
Beginner
Find the Distance Between
Point 1
(2, 3)
(x₁, y₁)
Point 2
(8, 11)
(x₂, y₂)
x₁
2
y₁
3
x₂
8
y₂
11
Distance Formula
d =
√
(x₂ − x₁)2
+
(y₂ − y₁)2
Keep Point 1 together and Point 2 together. Substitute each coordinate into the matching variable.
1
Substitute x
2
Substitute y
3
Subtract
4
Square & Add
5
Distance
Step 1
Substitute the x-coordinates
The first squared difference is x₂ − x₁. Use the labeled coordinate values above.
Current Problem
(x₁, y₁) =
(2, 3)
(x₂, y₂) =
(8, 11)
Your Work
Fill in x₂ and x₁.
Step 1 of 5
d =
√
(
x₂
−
x₁
)2
+
(
y₂
−
y₁
)2
x₂ comes from Point 2 and x₁ comes from Point 1.
Coordinate Plane
See where the two points lie on the coordinate plane.
Distance: —
Point 1
Point 2
Right Triangle
Horizontal Distance
—
Vertical Distance
—
Final Distance
—
Start by substituting x₂ and x₁ into the first
squared difference.
Hint
Point 2 is (8, 11), so x₂ is 8. Point 1 is (2, 3), so x₁ is 2.
Distance Formula Reference
Distance Formula
d =
√
(x₂ − x₁)2
+
(y₂ − y₁)2
Subtract the x-coordinates to find the horizontal change between the points.
Subtract the y-coordinates to find the vertical change between the points.
The coordinate differences form the legs of a right triangle. The distance is its hypotenuse.
Substitution checklist
x₂ goes first in the x difference
x₁ goes second in the x difference
y₂ goes first in the y difference
y₁ goes second in the y difference
Watch negative coordinates: if x₁ = −3, then substituting x₁ into x₂ − x₁ creates subtraction of a negative, such as 5 − (−3).
Problem Complete
d = 10
The horizontal difference is 6 and the vertical difference is 8, so the distance is √(6² + 8²) = √100 = 10.
Problems Solved
0
Correct Steps
0
Incorrect Attempts
0
Current Streak
0
Practice Round Complete
Great work!
You completed all 10 distance formula problems.
Problems Solved
0
Correct Steps
0
Incorrect Attempts
0
Step Accuracy
0%
Label the Coordinates
Write the first point as (x₁, y₁) and the second point as (x₂, y₂) before substituting.
Substitute Carefully
Match every coordinate to the correct variable, especially when one or more coordinates are negative.
Think Pythagorean Theorem
The horizontal and vertical coordinate differences create the legs of a right triangle whose hypotenuse is the distance.
Complete Example
Find the distance from (2, 3) to (8, 11)
Label the coordinates first, then substitute them into the formula in the same order.
(x₁, y₁) = (2, 3)
(x₂, y₂) = (8, 11)
d =
√
(x₂ − x₁)2
+
(y₂ − y₁)2
1
Substitute x
(8 − 2)²
2
Substitute y
(8 − 2)² + (11 − 3)²
3
Simplify the differences
6² + 8²
4
Square and add
√
36 + 64
=
√
100
5
Find the distance
d = 10
The two points are 10 units apart.
Practice the Distance Formula Step by Step
The distance formula finds the straight-line distance between two points on the coordinate plane. It uses the differences between the x-coordinates and y-coordinates to create the side lengths of a right triangle.
This interactive distance formula practice tool emphasizes substitution rather than simply asking for a final answer. Each problem labels the two coordinate pairs as (x₁, y₁) and (x₂, y₂), then guides students through the formula one part at a time.
How to Use the Distance Formula
Begin by labeling the coordinates. The first point provides x₁ and y₁, while the second point provides x₂ and y₂. Substitute those values into d = √ (x₂ − x₁)2 + (y₂ − y₁)2 .
Substitute the x-Coordinates
The first difference in the formula uses x₂ − x₁. Students should identify the x-coordinate from Point 2 first and the x-coordinate from Point 1 second. Keeping the coordinate labels visible helps prevent accidental swapping of x- and y-values.
Substitute the y-Coordinates
The second difference uses y₂ − y₁. Once both pairs of coordinates have been substituted, simplify the arithmetic inside each set of parentheses before squaring.
Distance Formula With Negative Coordinates
Negative coordinates require extra attention during substitution. If a coordinate being subtracted is negative, the expression may contain subtraction of a negative number. For example, 5 − (−3) simplifies to 8.
Simplifying Square Roots
Not every distance is a whole number. Some problems produce square roots that should remain exact, and others produce radicals that can be simplified. Intermediate and advanced practice can include both perfect-square and non-perfect-square distances.
Why the Distance Formula Works
The distance formula is based on the Pythagorean theorem. The difference between the x-coordinates gives the horizontal leg of a right triangle, while the difference between the y-coordinates gives the vertical leg. The distance between the points is the hypotenuse.
Three Difficulty Levels
Beginner practice focuses on manageable integer coordinates and distances that simplify cleanly. Intermediate practice introduces all four quadrants, subtraction of negative coordinates, and non-perfect square distances. Advanced practice includes more challenging coordinate combinations and radical simplification.
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