What you'll learn
Pythagoras' theorem relates the sides of a right-angled triangle and lets you find a missing side. In this guide you will learn the theorem, how to find the hypotenuse and shorter sides, how to apply it to problems in 2D, how to extend it to 3D, and how to recognise when to use it. This is a core geometry and measurement skill.
Key terms and definitions
Pythagoras' theorem — a² + b² = c², relating the sides of a right-angled triangle.
Hypotenuse — the longest side, opposite the right angle.
Right-angled triangle — a triangle with one 90° angle.
Shorter sides — the two sides that form the right angle.
Surd — a root left in exact form (e.g. √13).
Core concepts
The theorem
Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:
a² + b² = c²
where c is the hypotenuse (the longest side, opposite the right angle) and a and b are the two shorter sides. It only works for right-angled triangles.
Finding the hypotenuse
To find the hypotenuse, add the squares of the two shorter sides and take the square root: c = √(a² + b²). For example, with sides 3 and 4: c = √(9 + 16) = √25 = 5.
Finding a shorter side
To find a shorter side, subtract: rearrange to a² = c² − b², then square root. For example, if the hypotenuse is 13 and one side is 5: a = √(13² − 5²) = √(169 − 25) = √144 = 12. Remember to subtract when finding a shorter side, not add.
Applying in 2D problems
Many problems involve right-angled triangles hidden in shapes — ladders against walls, diagonals of rectangles, distances on a grid. Identify the right angle, label the hypotenuse, and apply the theorem. The distance between two points can be found by forming a right-angled triangle from the horizontal and vertical differences.
Extending to 3D
In 3D, you can find the longest diagonal of a cuboid using Pythagoras twice, or the 3D formula d = √(a² + b² + c²) for a cuboid with edges a, b, c. First find a diagonal across the base, then use it with the height to find the space diagonal.
Worked examples
Example 1: Hypotenuse
Find the hypotenuse of a right-angled triangle with sides 6 and 8.
c = √(6² + 8²) = √(36 + 64) = √100 = 10.
Example 2: Shorter side
The hypotenuse is 15 and one side is 9. Find the other side.
a = √(15² − 9²) = √(225 − 81) = √144 = 12.
Example 3: 3D diagonal
Find the space diagonal of a cuboid with edges 2, 3 and 6.
d = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7.
Common mistakes and how to avoid them
Adding when finding a shorter side. Subtract the squares to find a shorter side.
Misidentifying the hypotenuse. It is the longest side, opposite the right angle.
Using it on non-right-angled triangles. Pythagoras only works with a right angle (use the cosine rule otherwise).
Forgetting to square root. After a² + b² you must take the square root for the side.
Rounding too early. Keep exact values (surds) until the end if needed.
Exam technique for Pythagoras' Theorem
Identify the right angle and hypotenuse first.
Add squares for the hypotenuse, subtract for a shorter side.
Square root at the end to find the length.
Form right-angled triangles within shapes and grids.
Use the 3D formula √(a² + b² + c²) for cuboid diagonals.
Quick revision summary
Pythagoras' theorem applies to right-angled triangles: a² + b² = c², where c is the hypotenuse (longest side, opposite the right angle). To find the hypotenuse, add the squares of the two shorter sides and square root (sides 3, 4 → c = 5). To find a shorter side, subtract: a² = c² − b², then square root (hypotenuse 13, side 5 → 12). In 2D problems, spot the right-angled triangle in shapes, ladders, diagonals and grids, using horizontal and vertical differences for distances. In 3D, find a cuboid's space diagonal with d = √(a² + b² + c²), or apply Pythagoras twice. The common errors are adding instead of subtracting for a shorter side, misidentifying the hypotenuse, using it on non-right-angled triangles, forgetting to square root, and rounding too early. Identify the hypotenuse, add or subtract squares appropriately, square root at the end, and extend to 3D with the cuboid formula.