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how to find hypotenuse

To find the hypotenuse of a right triangle, you almost always use the Pythagorean theorem : a2+b2=c2 ,a^2+b^2=c^2,a2+b2=c2, where aaa and bbb are the legs and ccc is the hypotenuse.

Quick Scoop

  • The hypotenuse is the longest side in a right triangle, always opposite the right angle.
  • Core formula: c=a2+b2 ,c=\sqrt{a^2+b^2},c=a2+b2​.
  • Works for any right triangle as long as you know the other two sides.

Step‑by‑step: basic method

  1. Identify the right angle and the side across from it – that side is the hypotenuse ccc.
  1. Measure or note the lengths of the other two sides, call them aaa and bbb.
  1. Square them: compute a2a^2a2 and b2b^2b2.
  1. Add the squares: a2+b2a^2+b^2a2+b2.
  1. Take the square root of that sum: c=a2+b2 ,c=\sqrt{a^2+b^2},c=a2+b2​.

Example: If the legs are 3 and 4,
c=32+42=9+16=25=5,c=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5c=32+42​=9+16​=25​=5.

Special case: isosceles right triangle

If it’s a right triangle with the two legs equal (like a square cut along a diagonal):

  • Let the leg length be sss.
  • The hypotenuse is c=s2c=s\sqrt{2}c=s2​.

Example: legs each 10 → hypotenuse =102≈14.14=10\sqrt{2}\approx 14.14=102​≈14.14.

Other ways (when you know an angle)

If you know one acute angle θ\theta θ and a side:

  • If you know the side adjacent to θ\theta θ: c=adjacentcos⁡θc=\dfrac{\text{adjacent}}{\cos\theta}c=cosθadjacent​.
  • If you know the side opposite θ\theta θ: c=oppositesin⁡θc=\dfrac{\text{opposite}}{\sin\theta}c=sinθopposite​.

These come from basic trigonometry relationships in right triangles.

Tiny real‑life story example

Imagine you’re setting a ladder to reach a roof 10 ft high.

If you place it so the wall, ground, and ladder form a right triangle, the ladder is the hypotenuse, and you can find its needed length with the same formulas above, using either the height and base distance or height and angle.

Mini HTML table of key formulas

html

<table>
  <tr>
    <th>What you know</th>
    <th>How to find hypotenuse</th>
  </tr>
  <tr>
    <td>Both legs a, b</td>
    <td>c = √(a² + b²)</td>
  </tr>
  <tr>
    <td>Isosceles right, leg s</td>
    <td>c = s√2</td>
  </tr>
  <tr>
    <td>Angle θ and adjacent side</td>
    <td>c = adjacent / cos(θ)</td>
  </tr>
  <tr>
    <td>Angle θ and opposite side</td>
    <td>c = opposite / sin(θ)</td>
  </tr>
</table>

TL;DR: For almost all basic problems, use c=a2+b2 ,c=\sqrt{a^2+b^2},c=a2+b2​ as long as the triangle is right‑angled and you know the other two sides.

Information gathered from public forums or data available on the internet and portrayed here.