Two points and the right angle between them make a triangle. The distance is its hypotenuse.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The legs are easy
The horizontal leg is the gap in x. The vertical leg is the gap in y.
Then apply Pythagoras
From (1, 2) to (4, 6): the legs are 3 and 4, so the distance is √(9 + 16) = 5.
Signs do not matter
The legs are squared, so a gap of −3 gives the same 9 as a gap of 3.
Distance is Pythagoras in disguise
The gap between two points on a grid is the hypotenuse of a right triangle whose legs are the horizontal and vertical differences. So the distance is √((x₂−x₁)² + (y₂−y₁)²).
The differences get squared, so signs vanish
It does not matter which point you subtract from which, because squaring removes the sign. This makes the distance formula more forgiving than the slope formula, where order does matter.
Draw the triangle
Sketching the horizontal and vertical legs turns the formula into a picture you can check. If your answer is shorter than either leg, something is wrong — the hypotenuse is always the longest side.
Leave the root exact
A distance of √13 is exact; 3.61 is rounded. Unless a decimal is asked for, keep the surd. It is more precise and avoids rounding errors if the distance feeds into a further calculation.
Step 2: Try It Yourself
Set the two legs and read the hypotenuse.
The hypotenuse is not given. It comes from 3² + 4² = 25, whose square root is 5.
Step 3: Watch an Example
One step at a time.
Watch Ana Measure a Diagonal
Ana finds the distance from (2, 1) to (8, 9).
- Step 1
The gap in x is 8 − 2 = 6.
Step 4: Your Turn
Practice makes it stick.
The Diagonal
Problem 1 of 2
Legs of 3 and 4. What is the distance?
The Longer One
Problem 2 of 2
From (0, 0) to (5, 12). What is the distance?
Find the Hypotenuse
1 of 4
Legs 6 and 8. Distance?
2 of 4
From (1, 1) to (4, 5). Distance?
3 of 4
Legs 9 and 12. Distance?
4 of 4
From (−2, 0) to (1, 4). Distance?
Step 5: Quick Check
Show what you know.
Question 1 of 1
From (0, 0) to (8, 15). Distance?
What You Learned
- Two points on a grid make a right triangle with the axes.
- The legs are the gaps in x and in y.
- The distance is the hypotenuse, found by Pythagoras.