Q&A: What are some Common Errors and Misconceptions about the Pythagorean Theorem?

Cross-posted from here.

Here’s a list of mistakes that I’ve seen students make.

Conceptual Mistakes

• Applying the Pythagorean Theorem on non-right triangles. (They may also think that the word "hypotenuse" means the longest side of *any* triangle.)
• Assuming that the missing side is always the square root of the sum of squares of the other sides -- even when that missing side is a leg, not the hypotenuse. (They think the unknown side is always $c$ in the formula $a^2 + b^2 = c^2.$)
• Writing their final answer with $\pm$ even though the side length cannot be negative. (They use $c = \pm \sqrt{a^2 + b^2}$ because they're used to solving quadratic equations with no domain constraints, e.g., $x^2 = 9 \Rightarrow x = \pm 3.$)

Mindless Mistakes in Algebra

• Forgetting to take the square root at the end (i.e., $c = a^2 + b^2$).
• Forgetting to square the quantities within the square root (i.e., $c = \sqrt{a + b}$)
• Using a product instead of addition (i.e., $c = \sqrt{ab}$)

Misconceptions About Algebra

• Distributing the square root (i.e., $c = \sqrt{a^2 + b^2}$ $= \sqrt{a^2} + \sqrt{b^2}$ $= a + b$)
• "Canceling out" exponents of all terms in an equation (i.e., $a^2 + b^2 = c^2$ $\Rightarrow a + b = c$). Note that while this is technically the same error as distributing the square root, most students who make this error won't understand that and will continue to make it even if they understand that they cannot distribute a square root.