Prove area of Square: "If you pack four squares (resp. n^2) of side s and area A,..."
I have trouble understanding why the area of square of side "a" is its side squared without assuming the sides to be 1 unit and area 1 squnits.
I found an answer.But I have trouble understanding it.Can anyone please explain it?
Without reference to a unit:
If you pack four squares (resp. [FONT=MathJax_Math-italic]n^[FONT=MathJax_Main]2[/FONT][/FONT]) of side [FONT=MathJax_Math-italic]s[/FONT] and area [FONT=MathJax_Math-italic]A[/FONT], you get a bigger square, with sides twice (resp. [FONT=MathJax_Math-italic]n[/FONT] times) longer. Likewise, you can divide a square into "[FONT=MathJax_Math-italic]m^2 Squares"[/FONT] with sides [FONT=MathJax_Math-italic]m[/FONT] times smaller, and the side [FONT=MathJax_Math-italic]n[/FONT][FONT=MathJax_Math-italic]s[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Math-italic]m[/FONT] corresponds to the area ([FONT=MathJax_Math-italic]n^[FONT=MathJax_Main]2)([FONT=MathJax_Math-italic]A)[FONT=MathJax_Main]/([FONT=MathJax_Math-italic]m^[FONT=MathJax_Main]2)[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT]This shows that the ratio [FONT=MathJax_Math-italic]A/[FONT=MathJax_Math-italic]s^[FONT=MathJax_Main]2[/FONT][/FONT] is the same for all squares. Hence it is a "universal" constant, which we can denote [FONT=MathJax_Main]Σ[/FONT]. This gives us the formula for the area of the square:[/FONT]
I have trouble understanding why the area of square of side "a" is its side squared without assuming the sides to be 1 unit and area 1 squnits.
I found an answer.But I have trouble understanding it.Can anyone please explain it?
Without reference to a unit:
If you pack four squares (resp. [FONT=MathJax_Math-italic]n^[FONT=MathJax_Main]2[/FONT][/FONT]) of side [FONT=MathJax_Math-italic]s[/FONT] and area [FONT=MathJax_Math-italic]A[/FONT], you get a bigger square, with sides twice (resp. [FONT=MathJax_Math-italic]n[/FONT] times) longer. Likewise, you can divide a square into "[FONT=MathJax_Math-italic]m^2 Squares"[/FONT] with sides [FONT=MathJax_Math-italic]m[/FONT] times smaller, and the side [FONT=MathJax_Math-italic]n[/FONT][FONT=MathJax_Math-italic]s[/FONT][FONT=MathJax_Main]/[/FONT][FONT=MathJax_Math-italic]m[/FONT] corresponds to the area ([FONT=MathJax_Math-italic]n^[FONT=MathJax_Main]2)([FONT=MathJax_Math-italic]A)[FONT=MathJax_Main]/([FONT=MathJax_Math-italic]m^[FONT=MathJax_Main]2)[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT]This shows that the ratio [FONT=MathJax_Math-italic]A/[FONT=MathJax_Math-italic]s^[FONT=MathJax_Main]2[/FONT][/FONT] is the same for all squares. Hence it is a "universal" constant, which we can denote [FONT=MathJax_Main]Σ[/FONT]. This gives us the formula for the area of the square:[/FONT]
[FONT=MathJax_Math-italic]A[FONT=MathJax_Math-italic]s[FONT=MathJax_Main]=[FONT=MathJax_Main]Σ([FONT=MathJax_Math-italic]s^[FONT=MathJax_Main]2)[FONT=MathJax_Main].[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT]
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