The Pythagorean theorem

A right triangle has one $90^\circ$ angle. The two sides that form that angle are the legs, $a$ and $b$. The side opposite the right angle is the hypotenuse, $c$ — always the longest side.

Arrange four copies of the triangle inside a large square with side $a + b$, so that a tilted square of side $c$ is left in the middle. Count the area of the large square in two ways:

Expand the left side: $a^2 + 2ab + b^2 = 2ab + c^2$. Subtract $2ab$ from both sides and you get $a^2 + b^2 = c^2$.

• Mistake 1 · Adding the sides instead of their squares: $6 + 8 = 14$.

  Why it happens

  The theorem is about areas of squares, not lengths.

  What to do instead

  Square first, add second, then take the square root: $\sqrt{36 + 64} = 10$.

• Mistake 2 · Putting the hypotenuse in the wrong place when finding a leg: $5^2 + 13^2 = b^2$.

  Why it happens

  $c$ must be the longest side, so $c^2$ is always the sum.

  What to do instead

  Write $\text{leg}^2 + \text{leg}^2 = \text{hypotenuse}^2$, then subtract: $b^2 = 13^2 - 5^2$.

• Mistake 3 · Stopping at $c^2 = 100$ and answering $100$.

  Why it happens

  $100$ is the area of the square, not the side.

  What to do instead

  Finish with the square root: $c = 10$.

• Mistake 4 · Using the theorem on a triangle with no right angle.

  Why it happens

  The relationship only holds for right triangles.

  What to do instead

  Check for the $90^\circ$ angle first.

• Linear functions→Lines, slopes and how a graph tells a story.
• Linear equations→Solve equations step by step — and know why each step works.
• Learn→All ALMA math guides.