What Is A Hypotenuse?
A hypotenuse is the longest side of a right triangle - a triangle that contains one 90° angle - and it is the side directly opposite that right angle. The other two sides, which form the right angle between them, are called the legs (sometimes the base and the perpendicular).
Only right triangles have a hypotenuse. An equilateral or an obtuse triangle has three sides, but none of them earns the name, because there is no right angle for a side to sit opposite. The word itself comes from the Greek hypoteinousa, meaning "stretching under" — the side that stretches across from corner to corner.
Here is the fact that makes the hypotenuse special: the side opposite the largest angle in any triangle is the longest side. In a right triangle the 90° angle is the largest angle possible, because the other two must share the remaining 90°. So the side across from it wins the length contest every time.
The right-angled triangle is where this all lives, and the length of the hypotenuse is locked in by the two legs through one formula.
What Is The Hypotenuse Formula?
The hypotenuse length comes straight from the Pythagorean theorem:
$$c^2 = a^2 + b^2$$
Solving for the hypotenuse $c$:
$$c = \sqrt{a^2 + b^2}$$
The symbols mean:
$c$ is the hypotenuse - the side you are usually solving for.
$a$ and $b$ are the two legs - the sides that form the right angle.
Where does it come from? The Pythagorean theorem says the area of the square built on the hypotenuse equals the combined area of the squares built on the two legs. Because $c^2$ is a sum of two positive quantities, $c$ must be larger than either leg on its own. That is the algebra behind "always the longest side."
If instead you know the hypotenuse and one leg, rearrange:
$$a = \sqrt{c^2 - b^2}$$
One multiplication convention throughout this article: we use $\cdot$ for any product, never a bare $\times$.
Examples of Hypotenuse
Six worked examples, easier to harder. The problem statement is bold; the steps are not.
Example 1
Find the hypotenuse of a right triangle with legs 3 and 4.
$$c = \sqrt{3^2 + 4^2}$$ $$c = \sqrt{9 + 16}$$ $$c = \sqrt{25}$$ $$c = 5$$
Final answer: the hypotenuse is 5. This 3-4-5 set is the most famous Pythagorean triple and worth memorising.
Example 2
A student is told the two legs are 6 and 8 and writes the hypotenuse as 6 + 8 = 14. Is that right?
The tempting move is to add the legs directly, because the hypotenuse "goes further," so it should be bigger. Adding gives 14.
Check it against the theorem:
$$c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$$
The real answer is 10, not 14. Adding the legs ignores that you square first, add, then take the root - the squaring is what keeps the hypotenuse shorter than the straight sum of the two legs. Whenever a problem hands you two legs, reach for $\sqrt{a^2 + b^2}$, never $a + b$.
Example 3
Find the hypotenuse when both legs are 5.
$$c = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50}$$ $$c = 5\sqrt{2} \approx 7.07$$
Final answer: $5\sqrt{2}$, about 7.07. A right triangle with two equal legs is an isosceles right triangle, and its hypotenuse is always a leg times $\sqrt{2}$.
Example 4
A ladder leans against a wall. Its foot is 5 m from the wall and it reaches 12 m up. How long is the ladder?
The ladder is the hypotenuse; the wall height and the ground distance are the legs.
$$c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$$
Final answer: the ladder is 13 m long. (5-12-13 is another clean Pythagorean triple.)
Example 5
The hypotenuse is 13 and one leg is 5. Find the other leg.
Now you rearrange the formula to solve for a leg:
$$b = \sqrt{c^2 - a^2}$$ $$b = \sqrt{13^2 - 5^2}$$ $$b = \sqrt{169 - 25}$$ $$b = \sqrt{144} = 12$$
Final answer: the missing leg is 12. Notice the subtraction - solving for a leg means the hypotenuse's square comes first and the known leg is taken away.
Example 6
Two towns sit at coordinates $A(1, 2)$ and $B(4, 6)$. How far apart are they in a straight line?
The straight-line gap is the hypotenuse of a right triangle whose legs are the horizontal and vertical separations.
Horizontal leg: $4 - 1 = 3$ Vertical leg: $6 - 2 = 4$
$$\text{distance} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$
Final answer: the towns are 5 units apart. This is exactly the distance formula - it is the hypotenuse rule wearing a coordinate-geometry coat.
Why the Hypotenuse Earns Its Name
The hypotenuse is not just a vocabulary word - it is the reason right triangles run so much of applied geometry.
Ask why humans needed it, and the answer is old and practical:
Surveying and building. Long before coordinates existed, builders needed a reliable way to make a true right angle in the field. Stretching a loop of rope knotted into 3-4-5 segments produces a perfect 90° corner every time. The hypotenuse rule is what makes that trick work.
Measuring the unreachable. You cannot lay a tape measure up the face of a cliff or across a river. But you can measure two sides on the ground and let the hypotenuse relation give you the third. That is the whole engine behind navigation and early astronomy.
The straight-line shortcut. Any time you ask "how far apart are two points," you are asking for a hypotenuse. GPS, screen graphics, and collision detection in games all lean on it.
The idea is far older than its name. Clay tablets show that Babylonian mathematicians worked with these side relationships more than a thousand years before Pythagoras, using them to lay out fields and buildings. The relationship was discovered because people needed it, not because someone found it elegant on paper.
[INTERACTIVE: A draggable right triangle. The user drags the two leg endpoints; the legs $a$ and $b$ update live, and the hypotenuse $c$ recomputes as $\sqrt{a^2+b^2}$, shown both as a length and as the square-area picture (three squares built on the three sides, areas updating). A toggle shows the two leg-squares fitting into the hypotenuse-square.]
The Mistakes Students Make Most Often
Three failure modes cover almost every wrong hypotenuse answer.
Mistake 1: Adding the legs instead of using the theorem
Where it slips in: the very first time a student sees two leg lengths and is asked for the hypotenuse.
Don't do this: write $c = a + b$, for example $c = 6 + 8 = 14$.
The correct way: square, add, then root: $c = \sqrt{6^2 + 8^2} = 10$. The first instinct here is almost always to add the two legs, and the squaring step is exactly what gets dropped - it is what keeps the hypotenuse shorter than the straight sum of the sides.
Mistake 2: Solving for a leg with addition instead of subtraction
Where it slips in: problems that give you the hypotenuse and one leg and ask for the other leg (the rusher pattern - grab the formula that's memorised without checking direction).
Don't do this: write $b = \sqrt{c^2 + a^2}$ when the hypotenuse is already known.
The correct way: when the hypotenuse is given, subtract: $b = \sqrt{c^2 - a^2}$. The hypotenuse is the biggest square, so the known leg's square comes off it.
Mistake 3: Calling any long side a hypotenuse
Where it slips in: triangles that are not right triangles at all (the memorizer who learned "hypotenuse = longest side" but skipped the "right triangle" part).
Don't do this: label the longest side of an equilateral or obtuse triangle "the hypotenuse."
The correct way: first confirm there is a 90° angle. No right angle means no hypotenuse - only right triangles have one.
Conclusion
The hypotenuse is the side of a right triangle opposite the 90° angle, and it is always the longest side.
Find it with the Pythagorean theorem: $c = \sqrt{a^2 + b^2}$.
To find a missing leg when the hypotenuse is known, subtract: $b = \sqrt{c^2 - a^2}$.
Never add the legs directly - square, add, then take the root.
Only right triangles have a hypotenuse; always confirm the right angle first.
To take the hypotenuse further with a teacher, explore Bhanzu's geometry tutor, a high school math tutor for exam-level triangle work, or math classes online for structured practice.
Practice these problems to solidify your understanding: (1) legs 9 and 12; (2) legs 7 and 7; (3) hypotenuse 25, one leg 7 - find the other. Answer to Question 1: 15. Answer to Question 2: $7\sqrt{2} \approx 9.90$. Answer to Question 3: 24. If you get stuck on the leg-finding problem, come back to Example 5. Want a live trainer to walk through more triangle problems? Book a free demo class.
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