Geometry · GCSE Maths
Pythagoras
Use a² + b² = c² on right-angled triangles to find a missing side, including in 3D and in reverse to test for a right angle.
Pythagoras only cares about the right angle: the two shorter sides square and add; the hypotenuse squares alone.
The important bits
What you need to know
- 1
In a right-angled triangle, a² + b² = c², where c is the hypotenuse, the side opposite the right angle.
- 2
To find the hypotenuse, square, add, square-root. To find a shorter side, square, subtract, square-root.
- 3
Always identify the hypotenuse first. Using the longest given side as a when it is actually c is the standard error.
- 4
Leave answers as simplified surds when the question asks for exact values: √12 = 2√3.
- 5
The converse: if a² + b² = c² for three lengths, the triangle is right-angled at the vertex between a and b.
- 6
3D Pythagoras: find a face diagonal first, then use that diagonal with the third dimension as a new right-angled triangle.
- 7
Isosceles right-angled triangles and 30–60–90 triangles have exact side ratios you can use instead of a calculator.
- 8
Pythagoras and trigonometry often sit in the same question: find one side with Pythagoras, then an angle with SOHCAHTOA.
Go deeper
Subtract when you want a shorter side
Students remember “square, add, square-root” and then apply it even when the unknown is not the hypotenuse. If the hypotenuse is 13 and one side is 5, the other side is √(13² − 5²) = √(169 − 25) = √144 = 12, the familiar 5–12–13 triple. Adding would invent a side longer than the hypotenuse, which cannot happen in a right-angled triangle. A size check catches it: each shorter side must be less than c. Write c² = a² + b² and rearrange with the balance method rather than reciting a slogan. The equation tells you whether to add or subtract. Sketch the triangle and mark the right angle; if you cannot see which side is c, you are not ready to square anything.
Go deeper
Two triangles make a 3D length
A space diagonal of a cuboid does not sit on a face, so you cannot read all three edges as a single flat triangle. First find the diagonal of the base: if the base is 3 by 4, that diagonal is 5. Then the 5 and the height 12 form a second right-angled triangle whose hypotenuse is the space diagonal, 13. The same idea finds the height of a square-based pyramid: the base diagonal, halved, with the slant edge. Write the two triangles separately and name the shared length. Examiners look for that intermediate value; jumping from three edges to a single square-root often drops the method mark even when the final number is right. Keep exact surds until the last line if no rounding is requested.
See the idea in action
A right-angled triangle has shorter sides 7 cm and 24 cm. The hypotenuse is √(7² + 24²) = √(49 + 576) = √625 = 25 cm. If instead the hypotenuse is 25 cm and one side is 7 cm, the remaining side is √(625 − 49) = √576 = 24 cm.
Exam technique
Turn knowledge into marks
Write a² + b² = c² with the numbers under the letters before you calculate. If c is known, the next line should show a subtraction, not an addition.
Common mistakes
Do not give these marks away
- 01
Adding the squares when the unknown is a shorter side, producing a length longer than the hypotenuse.
- 02
Forgetting to square-root after adding or subtracting, and offering 625 as a length in centimetres.
- 03
Using Pythagoras on a triangle that has not been shown, or assumed, to be right-angled.
A right-angled triangle has shorter sides 5 cm and 12 cm. The hypotenuse is
A13 cm
B17 cm
C√17 cm
D60 cm
Show the answer
13 cm. 5² + 12² = 25 + 144 = 169 = 13². 17 cm would be the result of adding 5 and 12 without squaring.
Quick questions
If this is the bit you searched
Can I use Pythagoras if I only know one side and an angle?
Not by itself. One side and an angle is a trigonometry question. Pythagoras needs two sides in a right-angled triangle to find the third.
What are Pythagorean triples?
Whole-number sides that satisfy a² + b² = c², such as 3–4–5, 5–12–13 and 7–24–25, including their multiples 6–8–10 and 9–12–15.