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I'm puzzled about why the formula for area of a square is side squared, but for a rectangle it's length times height. Why wouldn't it be for a square, side times side?

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    $\begingroup$ It's the smae formula, except width = height = side for squares! $\endgroup$ – Bernard Oct 8 '17 at 16:31
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    $\begingroup$ In other words, side $\times$ side = side${}^2$. $\endgroup$ – J.-E. Pin Oct 8 '17 at 16:32
  • $\begingroup$ Actually you can get the area of the square also using the formula for the rhombus $$\frac{diagonal\times diagonal }{2}$$ $\endgroup$ – Raffaele Oct 8 '17 at 16:38
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The reason is simple and follows a common tradition in human communication far away from mathematics: We don't say "I ate an apple and an apple", though nothing factually or grammatically wrong. We prefer to say "I ate two apples"

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"Side times side" is equal to "side squared". So they are the same. $$\text{side }\times\text{ side} = \text{side}^2$$

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I am writing this to answers because I can't write it to the comments Assume that you have a square with side (say) $a$ units long, the area of this square (as we know it) is $a^2$;enter image description here

And then assume that you have pasted two squares horizontally;enter image description here

Now you have o add the area of these two squares $a^2+a^2=2a^2$ which we already knew from $side\times side$, this is a little demonstration of the fact that you can make two squares out of rectangles and add their areas together to prove that the area of a rectangle $=\text{side}\times\text{side}$. And my apologies because I am a beginner in Geogebra.

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