Distance between two points
Turn horizontal and vertical changes into a right triangle and calculate its hypotenuse.
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Subtract signed coordinates.
- Use Pythagoras' theorem and square roots.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is √(6² + 8²)?
What you will learn
- Find a line segment's length from two coordinates.
- Give exact or appropriately rounded lengths.
- Distinguish direct distance from a route along grid lines.
Build the right triangle
The straight segment between two points can be the hypotenuse of a right triangle whose other sides run horizontally and vertically. Their lengths are the absolute coordinate changes. For A = (−3, −2) and B = (3, 6), those lengths are six and eight, so AB² = 6² + 8² = 100 and AB = 10.
The formula is d = √[(x₂ − x₁)² + (y₂ − y₁)²]. Squaring makes either direction's changes positive. Subtract first, square each entire difference, add the squares, then take the square root. The result is a length, so it cannot be negative.
Exact form and scale
If the changes are two and three, the distance is √13. This is an exact answer; 3.61 is a rounded approximation. Preserve the square root until a question requests an approximation, and round only the final result. If the coordinates measure centimetres, the segment's length is in centimetres, not square centimetres.
An ordinary Cartesian diagram for distance uses the same physical unit in both directions. If a contextual graph has different quantities or scale units on its axes, applying a spatial distance formula to its displayed coordinate values may have no physical meaning. For a map whose coordinates both measure kilometres, the formula gives a distance in kilometres.
Compare direct and stepped routes
For changes six and eight, a route that goes horizontally then vertically is 6 + 8 = 14 units. The direct segment is ten units. Adding the two differences finds a stepped route length, not the hypotenuse. A direct distance may not be a usable route if a map shows walls, roads or other constraints.
For horizontal or vertical segments, one coordinate change is zero. The same formula then reduces to the absolute non-zero change. You can check whether a calculated length is plausible: it must be at least as large as either component and no larger than their sum.
Pause and explain
The changes are 2 and 3. What is the exact distance?
Worked example
Find the exact length from (−3, −2) to (3, 6).
Show the worked solution
- The coordinate changes are 6 horizontally and 8 vertically.
- Pythagoras gives d² = 36 + 64 = 100.
- Use the positive square root: d = 10 coordinate units.
Answer 10 units
Try a topic test
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the distance from (−1, 2) to (5, 10).
Give me a hint
Calculate the two changes before squaring.
Compare my reasoning
- Horizontal change is 6; vertical change is 8.
- d² = 6² + 8² = 100.
- d = 10 units, using the positive root.
10 units
Look for these in your work
- Used coordinate differences.
- Took the square root at the end.
Find the distance from (2, −4) to (−1, 0).
Give me a hint
The negative horizontal change will be squared.
Compare my reasoning
- The changes are −3 and 4.
- d² = (−3)² + 4² = 9 + 16 = 25.
- The distance is √25 = 5 units.
5 units
Look for these in your work
- Squared the complete signed difference.
- Returned a non-negative length.
On a map one coordinate unit represents 200 m. A = (−2, 1) and B = (4, 9). Compare their direct distance with a route travelling horizontally then vertically.
Give me a hint
Find both lengths in coordinate units before applying the map scale.
Compare my reasoning
- The changes are 6 and 8; the direct distance is √100 = 10 units.
- The stepped route is 6 + 8 = 14 units.
- These are 2000 m and 2800 m respectively, a difference of 800 m.
Direct 2 km; stepped 2.8 km; difference 0.8 km
Look for these in your work
- Distinguished the hypotenuse from the sum of components.
- Applied the map scale and converted units.
Common mistakes
- Adding horizontal and vertical changes for direct distance.
- Forgetting to take the square root.
- Rounding each intermediate calculation.
What can you explain now?
Give the exact distance when the coordinate changes are 1 and 2.
Compare with the explanation
√5 units
The sum of squares is 1 + 4 = 5; its square root is the exact length.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, draw a right triangle for a new pair of points and explain why the segment's length is positive even when a change is negative.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.