Why is it, if a circle of string is shaped differently, the interior area changes?

Question

Ok...A square of string - 4x4 ...area = 16.
so if it is shaped into a circle with a circumference of 16 (4x4) then the interior area changes!
Damn it - I know it's a simple thing, but I just cant get my head around it!

Answer 1

Imagine the square of string:



What will you do to make it a circle? You would have to bend the sides to form an arc like the way you would bend an archer's bow. For it to become a circle, you have to bend the four sides equally so that the corners will no longer be present.



So your new circle now will have an area bigger than the square. If you overlap the 2 figures together, you would have a big circle and inside it is a square whose 4 corners touch 4 equidistant points on the circle.



Area of the circle: 64/pi OR 20.3718

Answer 2

If you have a string - meaning a fixed circumference - then the largest area it can cover is a circle because all points are equally distant from the centre. As you move some points further out from the centre, others must move in closer since the string is of fixed length. Think of what happens when you stretch the string out as far as it can go - you end up effectively with two sides equal to half the length of the string and two others with effectively zero length - here the area is zero - the theoretical minimum area the string can cover. The more you make the sides equal, the more area you cover - until you get the other extreme - the circle

Answer 3

This is because you are cutting off all of the corners. Remember that when a square is 4 x 4 is is so on all four sides. Draw a 4 x 4 square then draw a circle inside the square where the circle only touches the square on the center of each side. That circle has a height and witdth or diameter of 4 same as the square but notice it no longer as the areas of each corner like the square does.

Answer 4

The perimeter of the string will stay the same but the inside area changes as the shape changes.



You could move the string so it formed a long rectangle that touched on the inside so that it had no interior area -

Answer 5

The fun part is to find the area of the missing corners when you make a circle out of the square.

Answer 6

Think of a box with the ends pushed out (4x4). Now push it sideways to form a parallelogram - keep pushing - the earea diminshes until you push it flat when the area approaches zero.



Perimeter on it's own bears no relationship to internal area

Answer 7

square have shape edges, which are like they r being squeezed. circles r fat squares. so, intuitively, their areas r larger, althogh they have tha same circumference.



Mathematically, u can work this one out, probably, with the famous fact that the hypertenus of a triangle h=sqrt(x^2+y^2). U can play with this yourself and if I have time later, I'll figure it out and describe it here.

Answer 8

a square and a circle are not the same shape.