How to find the hypotenuse of a right triangle
The hypotenuse is opposite the right angle and is the longest side of a right triangle. You can calculate it when the two perpendicular side lengths are known.
What the formula is saying
Pythagoras states a² + b² = c². To find c rather than c², square the legs, add the squares, then take the square root.
Read the symbols in plain language
- a
- First perpendicular sidem
- b
- Second perpendicular sidem
Sort out the units first
Both legs use the same length unit. Intermediate squares are m²; the square root returns to m. Do not round a square root too early.
Let’s solve one together
Read the given values, follow each operation, then check what the result means.
- a · First perpendicular side
- 9 m
- b · Second perpendicular side
- 12 m
Square the first leg
Squaring means multiplying a length by itself, not multiplying it by two.
(9)^2 = 81 m²Add the square of the second leg
The theorem adds squared lengths, not the lengths themselves.
(81) + (12)^2 = 225 m²Undo the square
The square root gives the actual hypotenuse length.
√((225)) = 15 m
Does this worked answer make sense?
The hypotenuse must be longer than each leg but shorter than their sum. 15 is between 12 and 21 in the worked case.
Now try your own values
Change a value or its unit. The same method will show your calculation, step by step.
Results update only when you calculate. The lesson example above stays unchanged.
Your turn — check your understanding
Use these new values. Work it out first, then check your answer.
- a · First perpendicular side
- 7 m
- b · Second perpendicular side
- 24 m
Find: the hypotenuse of a right triangle
A hint, not the answer
Pythagoras states a² + b² = c². To find c rather than c², square the legs, add the squares, then take the square root.
Both legs use the same length unit. Intermediate squares are m²; the square root returns to m. Do not round a square root too early.
Show the full practice solution
Compare the steps with your work; revealing a solution does not mark the lesson complete.
Square the first leg
Squaring means multiplying a length by itself, not multiplying it by two.
(7)^2 = 49 m²Add the square of the second leg
The theorem adds squared lengths, not the lengths themselves.
(49) + (24)^2 = 625 m²Undo the square
The square root gives the actual hypotenuse length.
√((625)) = 25 m
Avoid the common trap
9 + 12 = 21 is not the hypotenuse. This relation requires a right angle; it is not a universal triangle formula.
When this method applies — and when it does not
Euclidean right triangle with two positive perpendicular legs. To find a missing leg, rearrange the equation and subtract squares instead.
One idea understood. Keep going.
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Sources & further reading
References open in a new tab and explain the underlying principles. The teaching text and examples here are SimpleFlick’s own; the source organisations have not endorsed this calculator.
Lesson updated: · Worked examples checked against the implemented formula; not an independent engineering certification.