Key takeaway: The hypotenuse is the side of a right triangle that lies directly across from the right angle, and it is always the longest of the triangle’s three sides.
Grade 88.G.B.6Geometry and Measurement
Definition of Hypotenuse
The hypotenuse is the one side of a right triangle that does not touch the right angle. It stretches from the end of one leg to the end of the other, like the sloped edge of a wheelchair ramp or the diagonal of a rectangular door.
Because the right angle is the largest angle of the triangle, the side opposite to it is the longest. No matter how you rotate or flip the triangle, it stays the side facing the corner marked with a small square. Only right triangles have one.
Formula and notation
Letter \(c\) is the usual name for it, with legs \(a\) and \(b\):
\[c^{2} = a^{2} + b^{2}\]
In a triangle \(ABC\) with the right angle at \(C\), the side to look for is \(\overline{AB}\), the side whose letters are not the right-angle vertex.
Key facts
- It faces the right angle and never touches it.
- It is longer than each leg, so its length is less than the legs’ sum.
- Its direction on the page depends on how the triangle is turned.
- Its length comes from the Pythagorean theorem when both legs are known.
Hypotenuse example
A right triangle has one leg of 15 inches and a hypotenuse of 39 inches. Find the other leg.
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Put the longest side in \(c\): \(15^{2} + b^{2} = 39^{2}\).
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Compute: \(225 + b^{2} = 1521\), so \(b^{2} = 1296\).
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Square root: \(b = \sqrt{1296} = 36\).
The other leg is 36 inches, shorter than 39 inches, as it must be.
Common mistake
Watch out
Given a triangle drawn with the right angle at the top, Omar calls the bottom side the hypotenuse because it is the base and looks long.
Why it is wrong
Position on the page does not decide it. The hypotenuse is the side across from the right angle, and the right angle here is at the top, so the slanted side opposite it is the one you want. Find the small square marker first, then pick the side that does not touch it.
Related terms and lessons
Browse the glossary: A–Z list · Grade 8 vocabulary · Geometry and Measurement