Our teacher asked us to find and prove the scaling of a polygon's area when you scale it by a factor of a. I've searched for a long while, and I've found answers saying that it's a², but there is no proof for it. All I've found so far is a mention of this saying that it's hard to prove, or visual interpretations using a rectangle/triangle, but no general n-gon. I have no clue where to start since there is not a general formula for any n-gon. Could anyone help me out?
Let X be a square with side length 3 and A be 4. Scaling by a, we have a 12 x 12 square now. The area of this square is 12 x 12 = 144, \(4^2\) times bigger than the original square's area with 9. Thus, scaling by a increases the area by \(a^2\). (Note: If you still don't believe me, try this out with a different shape and different values for X and A.
I'm not sure what scaling means, but if it means what I think it means, consider the following.
Take a square, with sides S. The area is (S • S) = S2.
Scale it by a. Now the sides are (S • a) and the area is (S • a) • (S • a) = S2a2.
Make it a rectangle, scale by a, and the areas is (L • a) • (W • a) = LWa2
Try a circle of radius R, scale by a, and the area is π • (Ra)2 = πR2a2
Try watching this.
I ahve watch 3/4 of it and it looks really good.
Remember that If 2 figures are just different by a scaling factor, then they are similar figures.
Thanks. I did watch it, and I know why you quit 3/4 of the way through, LOL. :-)
ok thanks for responding. Did you get anything from it?
I quit 3/4 of the way through because I already know that stuff.
It was not the most exciting presentation but the content was good.
< Did you get anything from it? >
Yes, I did. It was really good. He confirmed what I had kinda figured out – I've taken higher math than that, but my courses never addressed polygon expansion – and he presented it in an understandable way. He repeats himself a lot, but that's better than not explaining enough. Thanks again for the link.
Thanks for the feedback. :)
You can take this one step further.
For any three dimensional object:
If you expand the lengths by a factor of y , [so a side of lenth 4 is epanded to length 4y]
then the area of any surface will become y^2 times bigger
and the volume will enlarge by y^3
This is useful for many physics applications too.