Alike but different dimensions

Hello fellow humans. My confusion is on the dimensions of objects. I noticed that if 3 shapes are made, A, B and C, all quadrilaterals. A is a 5x5, B is a 6x4 and C is a 8x2. The perimeter of all shapes are the same, 20, but their areas differ. Please indulge my inquisitiveness.

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Comments

Well it means that 2(l+b)=20 l+b=10. But this does not mean, that the area , lb, is fixed. l=8, b=2, gives lb=16. l=5,b=5 gives lb=25. l=6,b=4 gives lb=24.

Shourya Pandey - 8 years, 2 months ago

But why are their areas different in such a way that the area tends to be more towards the square shapes :)

Ibukunoluwa Abiodun - 8 years, 2 months ago

That is because: Suppose l>=b, without loss of generality. Say l=5+d. then b=5-d So lb=(5+d)(5-d)=25-d^2. So as l and b come closer to resemble a square, i.e., l=b=5, then d becomes smaller and smaller and the area increases.

Shourya Pandey - 8 years, 2 months ago

@Shourya Pandey – Another way is the A.M.-G.M. inequality, stating $\frac {l+b}{2}$ >= $\sqrt lb$ So that lb<=25, with equalty iff l=b=5, i.e., the figure is a square.

Shourya Pandey - 8 years, 2 months ago

@Shourya Pandey – Thanks a lot, now I have full light to my darkness, thanks to you. Man you are good :)

Ibukunoluwa Abiodun - 8 years, 2 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$