Hypotenuse
In short: The hypotenuse is the longest side of a right triangle, the one directly across from the \(90^{\circ}\) angle, and it is never one of the two legs.
8th GradeGeometry8.G.B.6
What is Hypotenuse?
Stand at the corner of a rectangular room and look at the opposite corner: the straight path across is the hypotenuse of the right triangle formed with two walls. It always faces the right angle.
Because the shortcut across is longer than either wall but shorter than walking along both, its length \(c\) always satisfies \(c > a\), \(c > b\) and \(c < a + b\). The Pythagorean theorem gives the exact value: \(c = \sqrt{a^2 + b^2}\).
How to spot it
- A diagram has a small square at one corner, and one side is drawn slanted across from it.
- The problem asks for a diagonal, a ramp length, a ladder length or a straight-line distance.
- Among three side lengths of a right triangle, you need to pick the biggest.
- Trigonometric ratios such as sine and cosine are written with this side in the denominator.
The method
- Find the right angle marker and trace the side that does not touch it; that side is the hypotenuse.
- If the two legs are known, compute \(c = \sqrt{a^2 + b^2}\).
- If a leg is unknown, use \(a = \sqrt{c^2 – b^2}\) with the hypotenuse as the larger number.
- Check that the answer exceeds both legs.
Hypotenuse: solved example
Problem
A wheelchair ramp rises 1.5 feet and covers a horizontal distance of 18 feet. How long is the sloped surface of the ramp, to the nearest hundredth of a foot?
Solution
The ramp surface is the side opposite the right angle, so it is the hypotenuse. The legs are 1.5 ft and 18 ft.
\(c^2 = 1.5^2 + 18^2 = 2.25 + 324 = 326.25\), so \(c = \sqrt{326.25} \approx 18.06\). The ramp surface is about 18.06 feet long.
Try it
Question 1
A right triangle has legs 20 and 21. Find its hypotenuse.
Show the answer
\(400 + 441 = 841\), and \(\sqrt{841} = 29\).
Question 2
A triangle has sides 9, 12 and 15. If it is a right triangle, which side is the hypotenuse, and does it check?
Show the answer
The longest side, 15. Check: \(81 + 144 = 225 = 15^2\), so yes.
Hypotenuse vs. Pythagorean Theorem
One is a segment: the longest side of a right triangle. The other is a rule linking all three sides, used to compute that segment or a leg. A name versus a tool.
Keep going
Lessons
Worksheets and quizzes
- The Pythagorean Theorem – Worksheet
- The Pythagorean Theorem – Quiz
- Right Triangle Trigonometry – Worksheet
- Right Triangle Trigonometry – Quiz
Related words
Whole dictionary: A–Z · 8th Grade · Geometry