What's the intuition behind Pythagoras' theorem? Today we learned about Pythagoras' theorem. Sadly, I can't understand the logic behind it.

$A^{2} + B^{2} = C^{2}$



$C^{2} = (5 \text{ cm})^2 + (7 \text{ cm})^2$ 
$C^{2} = 25 \text{ cm}^2 + 49 \text{ cm}^2$ 
$C^{2} = 74 \text{ cm}^2$
${x} = +\sqrt{74} \text{ cm}$

Why does the area of a square with a side of $5$&nbsp;cm + the area of a square with a side of $7$&nbsp;cm always equal to the missing side's length squared?

I asked my teacher but she's clueless and said Pythagoras' theorem had nothing to do with squares.

However, I know it does because this formula has to somehow make sense. Otherwise, it wouldn't exist.
Ponder this image and you will see why your intuition about what is going on is correct:
The accepted answer provides no explanation for the diagrams: 

Each diagram shows an outer square, $\color{red}{(a+b)}$ units on a side, divided into other figures. The diagrams suggest different expressions for the area of the outer square. Equating these expressions leads to the equation $a^2+b^2=c^2$.

Total area of $\color{red}{outer}$ square = Area not covered by triangles + Area covered by triangles

$\color{red}{(a+b)^2} = c^2 + 4(\dfrac1{2}ab)$  ________$\color{red}{(a+b)^2} = a^2 +b^2 + 4(\dfrac1{2}ab)$

$c^2 + 4(\dfrac1{2}ab)= a^2 +b^2 + 4(\dfrac1{2}ab)$

$\therefore c^2 = a^2+b^2$

source: Algebra Structure and Method Book $1$ by Brown, Dolciani et al.

Here is a visual aid to help the new learner move from one diagram to the other:
 

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