The 3D Pythagorean Theorem

[Kirsten’s note: Micah teaches Math, Physics, and test prep at Pacific Learning Academy. Below he’s reminding all of us students (old and young alike) that there’s more to Pythagoras’ theorem than we might remember!]

Most people are familiar with the traditional 2D Pythagorean theorem, which states that for any right triangle where a and b represent the legs and c represents the hypotenuse, the equivalence a² + b² = c² holds true. This allows us to determine any of the triangle’s side lengths if the other two are known.

The 2D Pythagorean theorem can be applied twice to calculate the longest diagonal inside a rectangular prism. If the base of the prism has dimensions x and y, and the diagonal along the base is represented by c, then x² + y² =  c². The longest diagonal in the solid, s, is the hypotenuse of the triangle formed by the sides c and the height of the solid, z. So we know that, c² + z² = s². From this you can see that it is possible to find the longest diagonal inside a rectangular prism using the  2D version of the theorem alone, but we can skip a step by using a slightly different equation.

There is a 3D analog to the Pythagorean Theorem that is both intuitive and useful. If we are attempting to calculate the longest diagonal that can fit inside a rectangular prism or cube we use the 3D Pythagorean Theorem. If the side lengths of the rectangular prism are x, y, and z, and the diagonal line connecting opposite corners of the rectangular prism is s, then the equivalence holds where x² + y² + z² = s²!

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This is a common problem in high school geometry and standardized tests. The 3D Pythagorean Theorem is also helpful for theoretical physicists as well as anyone moving large pieces of furniture! Try using this formula to find the length of the longest diagonal in a rectangular prism with dimensions 5 x 7 x 2 centimeters.

Image taken from: http://www.cosmosportal.org/articles/view/137291/#gen3

Answer to question = √ 78  cm³ ≈ 8.83 cm³

Pacific Learning Academy is a one-on-one high school offering single courses and dual enrollment, as well as full-time high school. We also offer tutoring in all subjects from 6th to 12th grade, including SAT/ACT diagnostic testing and prep, either in homes or local libraries across the Eastside (Issaquah, Sammamish, etc…). See more at www.PacificLearningAcademy.com.

March 30, 2012

This Post Has 3 Comments

  1. koobla says:

    the answer should propably be sqrt(78cm^2) => 8.83 cm (note the exponents)

  2. braden herman says:

    I’m an 8th grade student working on the pythagorean theorem and this is just what I needed. thanks!

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