How to Calculate the Diagonal of a Triangle

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If your teacher has asked you to calculate the diagonal of a triangle, she's already given you some valuable information. That phrasing tells you that you're dealing with a right triangle, where two sides are perpendicular to each other (or to say it another way, they form a right triangle) and only one side is left to be "diagonal" to the others. That diagonal is called the hypotenuse, and you can find its length using the Pythagorean Theorem.

TL;DR (Too Long; Didn't Read)

To find the length of the diagonal (or hypotenuse) of a right triangle, substitute the lengths of the two perpendicular sides into the formula ​a2​ + ​b2​ = ​c2​, where ​a​ and ​b​ are the lengths of the perpendicular sides and ​c​ is the length of the hypotenuse. Then solve for ​c​.

Pythagoras' Theorem

The Pythagorean Theorem – sometimes also called Pythagoras' Theorem, after the Greek philosopher and mathematician who discovered it – states that if ​a​ and ​b​ are the lengths of the perpendicular sides of a right triangle and ​c​ is the length of the hypotenuse, then:

a^2 + b^2 = c^2

In real-world terms, this means that if you know the length of any two sides of a right triangle, you can use that information to find out the length of the missing side. Note that this only works for right triangles.

Solving for the Hypotenuse

Assuming you know the lengths of the two non-diagonal sides of the triangle, you can substitute that information into the Pythagorean Theorem and then solve for ​c.

  1. Substitute Values for a and b

  2. Substitute the known values of ​a​ and ​b​ – the two perpendicular sides of the right triangle – into the Pythagorean Theorem. So if the two perpendicular sides of the triangle measure 3 and 4 units respectively, you'd have:

    3^2 + 4^2 = c^2
  3. Simplify the Equation

  4. Work the exponents (when possible – in this case you can) and simplify like terms. This gives you:

    9 + 16 = c^2

    Followed by:

    c^2 = 25
  5. Take the Square Root of Both Sides

  6. Take the square root of both sides, the final step in solving for ​c​. This gives you:

    c = \sqrt{25}= 5

    So the length of the diagonal, or hypotenuse, of this triangle is 5 units.


    • What if you know the length of the triangle's diagonal and one other side? You can use the same formula to solve for the length of the unknown side. Just substitute in the lengths of the sides you do know, isolate the remaining letter variable on one side of the equals sign, and then solve for that letter, which represents the length of the unknown side.


About the Author

Lisa studied mathematics at the University of Alaska, Anchorage, and spent several years tutoring high school and university students through scary -- but fun! -- math subjects like algebra and calculus.