The hypotenuse is the longest side of a right triangle. It is always across from the right angle, and the two shorter sides that form the right angle are the legs. In the Pythagorean theorem a2 + b2 = c2, the hypotenuse is c: in a 3-4-5 right triangle, it is the side of length 5.
In Spanish
hipotenusa
Also called
longest side of a right triangle, side c
How to say it
hy-POT-uh-noos
Grade level
6-8
Grade 8Algebra 1CCSS 8.G.B.6CCSS 8.G.B.7GEDSATACT
Hypotenuse in a picture
Find the square corner, then look straight across it: that side is the hypotenuse, however the triangle is turned. It is always the longest of the three sides.
Formal definition and notation
In a right triangle, the side opposite the 90° angle is the hypotenuse; the other two sides are the legs. If the legs have lengthsa and b and the hypotenuse has length c, the Pythagorean theorem states a2 + b2 = c2, so c = √a2 + b2. Because the right angle is the largest angle of the triangle, the hypotenuse is the longest side: c > a and c > b. Triangles without a right angle have no hypotenuse.
Symbol
Read it as
Meaning
c
c
the usual letter for the hypotenuse; a and b are the legs
AB
segment A B
the hypotenuse of right triangle ABC when the right angle is at C
To find a leg instead, subtract: a = √c2 − b2. For a 45-45-90 triangle with legs s, the hypotenuse is s√2.
Try it: drag the legs
Interactive
Drag the ends of the two legs, or use the arrow keys. The hypotenuse updates with c = √a2 + b2. Look for leg pairs that give a whole-number hypotenuse, and notice that c always beats the longer leg.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Hypotenuse examples, step by step
Example 1: Find the hypotenuse
Warm-up
A right triangle has legs 6 cm and 8 cm. How long is the hypotenuse?
Square the legs: 62 = 36 and 82 = 64.
Add: 36 + 64 = 100, so c2 = 100.
Take the square root: c = √100 = 10.
Answer: The hypotenuse is 10 cm.
Example 2: A hypotenuse that is not a whole number
Core
Find the hypotenuse of a right triangle with legs 5 and 7. Round to the nearest tenth.
c2 = 52 + 72 = 25 + 49 = 74.
c = √74. Since 82 = 64 and 92 = 81, c is between 8 and 9.
A calculator gives √74 ≈ 8.60.
Sense check: 8.6 is longer than the longer leg, 7, as a hypotenuse must be.
Answer: The hypotenuse is √74, about 8.6 units.
Example 3: The ladder problem
Test-style
A 13-foot ladder leans against a wall. Its foot is 5 feet from the base of the wall. How high up the wall does the ladder reach?
The wall and ground form the right angle, so the ladder, opposite it, is the hypotenuse: c = 13.
The ground distance is a leg, b = 5. The height is the other leg, a.
a2 + 52 = 132, so a2 = 169 − 25 = 144.
a = √144 = 12. Trap: adding 132 + 52 treats the ladder as a leg and gives about 13.9 feet, taller than the ladder itself.
Answer: The ladder reaches 12 feet up the wall.
Common mistakes
✗ MistakeA triangle has sides 6, 7 and 9, so its hypotenuse is 9.
✓ Instead62 + 72 = 85, not 81 = 92, so this is not a right triangle and it has no hypotenuse.
The hypotenuse exists only in a right triangle. Any triangle has a longest side, but only a right triangle’s longest side is called the hypotenuse.
✗ MistakeSides 9, 12 and 15: using c = 12, 92 + 152 = 122.
c in a2 + b2 = c2 must be the hypotenuse, the side across from the right angle, not whichever side is listed last.
✗ MistakeLegs 3 and 4, so the hypotenuse is 3 + 4 = 7.
✓ Insteadc = √9 + 16 = √25, so c = 5.
Square the legs, add, then take the square root. Adding the legs gives a length longer than the shortcut across, which is impossible.
✗ Mistakec2 = 64 + 225 = 289, so c = 289.
✓ Insteadc = √289 = 17
289 is c2, the area of the square on the hypotenuse. The last step is the square root.
Not to be confused with
Hypotenusevs.Leg
The legs are the two sides that form the right angle; the hypotenuse is the third side, across from it. In a 5-12-13 triangle the legs are 5 and 12 and the hypotenuse is 13.
The hypotenuse is a side. The Pythagorean theorem is the rule a2 + b2 = c2 that links its length to the legs.
Hypotenusevs.Longest side of any triangle
Every triangle has a longest side, opposite its largest angle. It is called the hypotenuse only when that angle is exactly 90°.
Quick check: 3 questions
Question 1 of 3
A right triangle has legs 9 and 12. How long is the hypotenuse?
Show answer and explanation
Answer: A.92 + 122 = 81 + 144 = 225, and √225 = 15. 21 adds the legs, 225 forgets the square root, and √63 subtracts the squares.
Question 2 of 3
In triangle ABC, angle C is 90°. Which side is the hypotenuse?
Show answer and explanation
Answer: C. The hypotenuse is across from the right angle, so it is the side that does not touch C: side AB. Sides AC and BC meet at C, so they are the legs.
Question 3 of 3
A right triangle has hypotenuse 10 and one leg 6. How long is the other leg?
Show answer and explanation
Answer: D.b2 = 102 − 62 = 100 − 36 = 64, so b = 8. √136 treats 10 as a leg, and 4 and 16 subtract or add the lengths without squaring.
Hypotenuse: frequently asked questions
Is the hypotenuse always the longest side?
Yes. In any triangle the longest side is across from the largest angle. In a right triangle the 90° angle is the largest, because the other two angles share the remaining 90°. So the side across from it, the hypotenuse, is always the longest. The Pythagorean theorem agrees: c2 = a2 + b2 is larger than a2 or b2 alone.
Does every triangle have a hypotenuse?
No. Only right triangles have one. An equilateral triangle with sides 4, 4, 4 or an obtuse triangle with sides 4, 5, 8 has a longest side, but no right angle, so none of its sides is a hypotenuse. To check whether a triangle with sides a ≤ b ≤ c is right, test whether a2 + b2 = c2.
How do you find the hypotenuse without a calculator?
Square both legs, add, and take the square root. For legs 8 and 15: 64 + 225 = 289 = 172, so the hypotenuse is 17. Memorizing a few Pythagorean triples (3-4-5, 5-12-13, 8-15-17, 7-24-25) and their multiples, such as 6-8-10, saves time on tests.
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