Pythagoras' theorem, explained with examples

By The STEM-Tutor team · Updated October 2026

Pythagoras' theorem links the three sides of any right-angled triangle. It is one of the most useful results in GCSE Maths and appears on both Foundation and Higher papers.

The theorem

In a right-angled triangle with shorter sides a and b and hypotenuse c,

a² + b² = c²

The hypotenuse is the longest side. It is always opposite the right angle. The theorem only works for right-angled triangles, so check for the little square marking the right angle before you use it.

A good way to picture it: draw a square on each side of the triangle. The area of the square on the hypotenuse equals the total area of the squares on the other two sides. For a 3, 4, 5 triangle, that is 9 + 16 = 25.

Finding the hypotenuse

To find the longest side, square, add, square root: square the two shorter sides, add them, then take the square root.

Example 1: the hypotenuse

A right-angled triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse.

• c² = 6² + 8² = 36 + 64 = 100.
• c = √100 = 10 cm.

Example 2: a decimal answer

Find the hypotenuse when the shorter sides are 5 cm and 7 cm, to 1 decimal place.

• c² = 25 + 49 = 74.
• c = √74 ≈ 8.6 cm.

Finding a shorter side

To find one of the shorter sides, square, subtract, square root: square the hypotenuse, subtract the square of the known short side, then take the square root. Always subtract the smaller square from the bigger one.

Example 3: a shorter side

The hypotenuse is 13 cm and one side is 5 cm. Find the other side.

• a² = 13² − 5² = 169 − 25 = 144.
• a = 12 cm.

Example 4: a ladder

A 6 m ladder leans against a wall with its foot 1.5 m from the wall. How high up the wall does it reach, to 2 decimal places?

• The ladder is the hypotenuse. h² = 6² − 1.5² = 36 − 2.25 = 33.75.
• h = √33.75 ≈ 5.81 m.

Quick check: is my answer sensible?

The hypotenuse must be longer than each of the other two sides, but shorter than their total. If you find a hypotenuse of 4 cm in a triangle with a side of 6 cm, you have subtracted when you should have added. If you find a shorter side that is longer than the hypotenuse, you have added when you should have subtracted. This check takes two seconds and catches the most common error.

Pythagorean triples

Some right-angled triangles have whole-number sides. These sets are called Pythagorean triples. The most common are 3, 4, 5, 5, 12, 13, 8, 15, 17 and 7, 24, 25. Any multiple of a triple is also a triple, so 6, 8, 10 and 9, 12, 15 work too. Spotting these saves time, especially on the non-calculator paper.

Pythagoras on the non-calculator paper

Questions on the non-calculator paper are usually designed so the numbers work out neatly, often using a Pythagorean triple or a multiple of one. If the answer is not a whole number, you may be asked to leave it as a surd, such as √74, or to simplify it, such as √50 = 5√2. Knowing your square numbers up to 15² = 225 makes these questions much faster, and helps you spot when an answer looks wrong.

Distance between two points

To find the distance between two points on a coordinate grid, draw a right-angled triangle with a horizontal side and a vertical side. The horizontal side is the difference in x-coordinates and the vertical side is the difference in y-coordinates.

Example 5: distance between points

Find the distance between A(1, 2) and B(7, 10).

• Horizontal: 7 − 1 = 6. Vertical: 10 − 2 = 8.
• Distance = √(36 + 64) = √100 = 10 units.

Proving a triangle is right-angled

The theorem also works in reverse. If the sides of a triangle satisfy a² + b² = c², the triangle must be right-angled. Exam questions sometimes give three lengths and ask whether the triangle has a right angle.

Example 6: checking for a right angle

A triangle has sides 9 cm, 12 cm and 16 cm. Is it right-angled?

• 9² + 12² = 81 + 144 = 225, but 16² = 256.
• 225 ≠ 256, so it is not right-angled. Since 225 < 256, the largest angle is greater than 90°.

Pythagoras in 3D (Higher)

In a cuboid, the longest diagonal runs from one corner to the opposite corner. Find it in two steps: first the diagonal across the base, then use that as one side of a vertical right-angled triangle. Or use the shortcut d² = l² + w² + h².

Example 7: the diagonal of a cuboid

A cuboid measures 3 cm by 4 cm by 12 cm. Find the length of its longest diagonal.

• Base diagonal: √(9 + 16) = 5 cm.
• Space diagonal: √(5² + 12²) = √169 = 13 cm. The shortcut gives √(9 + 16 + 144) = 13 too.

Common mistakes

• Using Pythagoras on a triangle that is not right-angled.
• Adding when finding a shorter side, or subtracting when finding the hypotenuse.
• Forgetting to take the square root at the end.
• Treating the wrong side as the hypotenuse; it is always opposite the right angle.
• Rounding the squared value before taking the square root.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Is Pythagoras on the Foundation paper?

Yes. Finding sides in 2D right-angled triangles is on both tiers. Pythagoras in 3D is usually a Higher topic.

Is the formula given in the exam?

For GCSE exams in 2025, 2026 and 2027, a formulae sheet is provided with the Maths papers, covering formulae students previously had to memorise. Check the sheet for your exam year. Learn the formula anyway: recalling it instantly saves time and avoids copying errors.

When do I use Pythagoras and when do I use trigonometry?

Use Pythagoras when you know two sides and want the third. Use trigonometry (SOHCAHTOA) when an angle is involved, either given or wanted.

Key takeaways

• a² + b² = c², where c is the hypotenuse.
• Hypotenuse: square, add, square root.
• Shorter side: square, subtract, square root.
• Learn the triples 3, 4, 5 and 5, 12, 13.
• In 3D, use two triangles or d² = l² + w² + h².

Related guides

• GCSE Maths trigonometry
• GCSE Maths surds
• GCSE Maths topics