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Distance Between Two Points

FoundationHigherAQAEdexcelOCR

This free Foundation and Higher GCSE Maths worksheet on distance between two points helps you revise finding the distance between two points. Questions build from straightforward to exam standard, with full worked answers in the mark scheme — ideal for revision or homework. Suitable for AQA, Edexcel and OCR. Use Pythagoras on the horizontal and vertical gaps.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

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Higher

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Topic overview

The distance between two points is found using Pythagoras' theorem. The horizontal and vertical gaps form the two shorter sides of a right-angled triangle, and the distance is the hypotenuse.

The formula is \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Because both differences are squared, the order of subtraction does not matter, which removes one common worry about signs.

Answers are often left in surd form when the question asks for an exact value. Rounding \(\sqrt{13}\) to \(3.61\) loses accuracy, so read carefully whether an exact or decimal answer is wanted.

Revision notes

Setting up the triangle

Find the horizontal difference and the vertical difference between the points. These are the legs of a right-angled triangle.

For \((1, 2)\) and \((4, 6)\): the differences are \(3\) and \(4\), which are the two shorter sides.

Applying Pythagoras

Square both differences, add them, then take the square root.

Here \(3^2 + 4^2 = 25\), so the distance is \(\sqrt{25} = 5\). Squaring removes any negatives, so subtraction order is irrelevant.

Exact answers in surd form

When the square root is not a whole number, leave it as a surd unless a decimal is requested.

For a difference of \(2\) and \(3\), the distance is \(\sqrt{13}\), which is exact. Simplify surds where possible, so \(\sqrt{8}\) becomes \(2\sqrt{2}\).

Key points

  • Distance uses Pythagoras' theorem.
  • \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).
  • The differences form the two shorter sides.
  • Squaring removes negative signs.
  • Leave answers as surds when an exact value is asked for.
  • Simplify surds where possible.

Worked examples

Example 1

Find the distance between \((2, 3)\) and \((8, 11)\).

Working

\[8 - 2 = 6 \text{ and } 11 - 3 = 8\]find the horizontal and vertical differences
\[6^2 + 8^2 = 100\]square and add
\[\sqrt{100} = 10\]take the square root

Example 2

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

Working

\[4 - 1 = 3 \text{ and } 3 - 1 = 2\]find both differences
\[3^2 + 2^2 = 13\]square and add
\[\sqrt{13}\]leave as a surd for an exact answer

Example 3

Find the distance between \((-2, 1)\) and \((2, 4)\).

Working

\[2 - (-2) = 4 \text{ and } 4 - 1 = 3\]find the differences, taking care with the negative
\[4^2 + 3^2 = 25\]square and add
\[\sqrt{25} = 5\]take the square root

Common mistakes

  • Forgetting to square the differences.

    Adding 3 and 4 gives 7, not 5. Both differences must be squared first.

  • Rounding when an exact answer is wanted.

    √13 is exact; 3.61 is not. Read the question.

  • Sign errors with negative coordinates.

    2 − (−2) is 4, not 0. Subtracting a negative adds.

  • Forgetting the square root.

    Stopping at 25 gives the square of the distance, not the distance.

Exam tips

  • Sketch the two points and the right-angled triangle if it helps.
  • Square both differences before adding.
  • Leave surds unsimplified only if they cannot be simplified.
  • Check the distance is longer than either individual difference.

Key terms

Hypotenuse
The longest side of a right-angled triangle.
Surd
A root that cannot be simplified to a whole number.
Exact value
An answer left in surd or fraction form.
Difference
The gap between two values.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.