Can two different shapes have different areas and perimeters, but have the same lengths and widths?

Question

This is in regards to squares and rectangles only?

I am writing a C++ program, and I need to know for one of my if statements. I originally thought that this could not happen, but my girlfriend's mother seems to think this condition can be true. So... can it?

Anonymous · Answer

Quote: " Can two different shapes have different areas and perimeters, but have the same lengths and widths? ..." ;

..... No not IF you are confined to geometries within the Euclidean Plane ( which I am sure was your intention when asking the question) ;

But it can happen that they WILL be different if you extend the geometries to include non-Euclidean Planes and, for example construct one "rectangle" on the surface of a sphere, and the other "rectangle" on the surface of a  flared trumpet "bell".

Most people are happy to accept that they can walk along the perimeter of a square on the ground simply by walking the same distances (say 30 metre) in each of the four directions: initially north and continue walking for 30 metre, then west and walk 30 metre, then likewise for south then east.

But suddenly, they decide it isn't really a square if they start, shall we say, (100 / π) metre away from the north pole, when that will cause them to walk half the distance towards the north pole, and the other half away from the north pole, to bring them back to the same latitude, that they they follow towards the west,while maintaining their original distance from the north pole, before starting to move south and further away from the north pole to a latitude further away from the north pole  before finally moving east with no chance of walking what they would happily call a "square route".

Johan · Answer

Consider a rectangle with width W and length L, and an ellipse with same width W and same length L....

(You're asking explicitly about "two different shapes". If they are both rectangles, then they aren't different shapes.)

(And please don't retort with "but a square is a different shape from a rectangle", because it isn't. A square is also a rectangle.)

? · Answer

A rectangle is completely specified by its length and width.  There are no other "degrees of freedom".  Two rectangles that have the same length and width are identical, and so have the same area and perimeter.

and anyway:

perimeter = 2 x (length + width)
area = length x width

So if the length and width are the same, how can the perimeter or area be different?

? · Answer

In case of squares and rectangles , answer is NO but in case of Parallelogram answer is YES

Can two different shapes have different areas and perimeters, but have the same lengths and widths?

4 Answers