Thread: LOGICAL connection between surface and area

I got a problem which I am pondering about for a while.

It's about a general relationship between volume and surface area. It could also be between a line and an area.

The thing is that it's possible that two objects can have the same volume but different surface area. I don't understand that logically. It's completely clear and fine for me in a mathematical way, but not in a logical way.

Let's consider an object with the volume x. Now this volume is enclosed, covered, trapped (whatever you want to call it) by a surface area. I can change the shape of the volume, and still the surface would enclose this volume - but the surface area can change during this.
The same with a plane. The plane is encompassed by lines. There are many shapes of the plane, it's still got the same area. But why would the lenght of the lines (circumference) change when just the shape changes, but not the area.

I mean, it's technically still the same area, just in another shape - or the same volume, just in another shape.

Now my question to solve this problem would be:
Is there a general relationship, applicable to all shapes of volumes and surfaces, which "links" volume and surface area (or circumference and area)?
Is there a concept which can be used to consider such things as volume and area in a way that my problem wouldn't exist?

Thanks in advance.

There are many shapes of the plane, it's still got the same area. But why would the lenght of the lines (circumference) change when just the shape changes, but not the area.

I think this is your problem narrowed down.

you can logically think of this problem in simple terms.

on a cartesian coordinate system, the area and perimeter are 2 different functions of the same variables (x,y)

so you can vary x and y (the shape) as you wish to enclose your set area, but the perimeter (which is a different funtion of x and y) will change.

Is there a general relationship, applicable to all shapes of volumes and surfaces, which "links" volume and surface area (or circumference and area)?

this can't be. From above we can see that area and perimeter are seperate functions, and any link between the two will depend on the shape, thus removing any general relationship for all shapes


(with circles it may seem only one variable exists but by defining the shape as a circle you remove a degree of freedom)

The thing is that it's possible that two objects can have the same volume but different surface area. I don't understand that logically. It's completely clear and fine for me in a mathematical way, but not in a logical way.

Let's say you have an object of size 8x8x1. Now you cut it and get four pieces of size 4x4x1. These pieces together have a higher surface area than the first object. It is clear where this new surface area came from - the cuts. Now you glue them together to get a cube. The cube has even lower surface area than the first object - it lost the area of the glued surfaces.