Pythagoras’ theorem.
Identify the hypotenuse, square the sides and uncover a missing length.
Use Pythagoras for right triangles · find shorter sides and diagonals · check right angles
The square on the longest side
In a right-angled triangle, one angle is exactly 90°. The side opposite it is the hypotenuse, always the longest side. Identify that side before using any relationship between the lengths.
Pythagoras says that the area of a square built on the hypotenuse equals the combined areas of squares on the two shorter sides. This is why the formula contains squares, not just a sum of the side lengths. It applies specifically to right-angled triangles.
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
The hypotenuse c is opposite the right angle and is always the longest side. Letters can change from one question to another; identify the side by its position.
Pause and explainCan Pythagoras be used directly on any triangle?
No. A right angle must be given or established first.
Find a hypotenuse or a shorter side
Finding the hypotenuse and finding a shorter side are reverse uses of the same equation. In both cases, label the hypotenuse first so the squared terms are placed correctly.
Add the shorter-side squares to find the hypotenuse square. To recover a shorter-side square, subtract the other shorter-side square from the hypotenuse square. The final square root turns a squared length back into a length; leaving the squared result is not a completed answer.
A right triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse.
c = √100 = 10 cm
The hypotenuse is 13 m and one shorter side is 5 m. Find the other side.
b = √144 = 12 m
Subtract when the unknown is a shorter side. Take the positive root because a length is positive.
Find the exact diagonal of a 4 cm by 7 cm rectangle.
√65 is exact. A calculator approximation is 8.06 cm to 3 significant figures.
Pause and explainA hypotenuse is 13 and one shorter side is 5. Which squares are subtracted?
13² − 5². Taking the square root of 169 − 25 gives the missing shorter side, 12.
Check the triangle and apply it in context
A practical picture may hide the right-angled triangle. A ladder against a vertical wall, a rectangle’s diagonal or a vertical drop can provide the required perpendicular sides.
Draw that triangle separately and transfer only its relevant lengths. After calculating, use the geometry to check your answer: the hypotenuse must be longest, and a diagonal must exceed either of its perpendicular component lengths. Keep exact roots where asked.
Do side lengths 5, 12 and 13 form a right triangle?
The longest side is 13. Compare 5² + 12² = 169 with 13² = 169. They are equal, so the triangle is right-angled by the converse of Pythagoras’ theorem.
A 10 m ladder reaches 8 m up a vertical wall. Find its horizontal distance from the wall.
The ladder is the hypotenuse; the wall and level ground meet at 90°.
These are useful right-triangle triples. Multiplying every side by the same positive factor preserves the right angle.
Pause and explainWhat quick check would reject a calculated hypotenuse smaller than a known leg?
The hypotenuse is the longest side of a right triangle, so such a result cannot be correct.
Your turn. Locate the right angle first.
Mark the right angle and circle the hypotenuse on every sketch. Then decide whether its square is being found by addition or a shorter side by subtraction.
Show the square-root step and include units. Before using Pythagoras in a word problem, explain where the right angle comes from.
Find the hypotenuse of a right triangle with shorter sides 9 cm and 12 cm.
Reveal worked answer
A right triangle has hypotenuse 17 m and one shorter side 8 m. Find the other side.
Reveal worked answer
Find the exact diagonal of a square of side 5 cm.
Reveal worked answer
Are sides 3, 4 and 6 the sides of a right triangle?