High School Why do shapes with the same area have different perimeters?

[SSG-E]

TL;DR

For example, Consider two shapes; a circle and rectangle.

Both these shapes have same area but the perimeter of circle is less than that of rectangle. Why?

 

[DrClaude]

Take a loop of string or a rubber band and play around with it. It will give you a feeling as to why a constant perimeter can result in different areas.

 

[phyzguy]

I don't know what kind of answer you are looking for. It's obvious from geometry that there are many possible shapes with the same area and different perimeters. If I make a very long skinny rectangle of a given area, I can make the perimeter as large as I want. I don't know what kind of answer to give to your question except, "that's the way the geometry of our universe works."

 

[PeroK]

Summary:: For example, Consider two shapes; a circle and rectangle.

Both these shapes have same area but the perimeter of circle is less than that of rectangle. Why?

Or, try to find an example of two different shapes that have the same area and the same perimeter.

 

[SSG-E]

Or, try to find an example of two different shapes that have the same area and the same perimeter.

none

 

[SSG-E]

I don't know what kind of answer you are looking for. It's obvious from geometry that there are many possible shapes with the same area and different perimeters. If I make a very long skinny rectangle of a given area, I can make the perimeter as large as I want. I don't know what kind of answer to give to your question except, "that's the way the geometry of our universe works."

I mean is it something related to the sides of the shapes?

 

[DrClaude]

I mean is it something related to the sides of the shapes?

The answer is trivially yes.

Start with a simple problem: consider a rectangle with sides of length $a$ and $b$. The area is given by $a \times b$ while the parameter is $2 (a+b)$. Do you see how a quantity that changes with the product of variables behaves differently than a quantity that changes with the sum of those variables?

 

[SSG-E]

The answer is trivially yes.

Start with a simple problem: consider a rectangle with sides of length $a$ and $b$. The area is given by $a \times b$ while the parameter is $2 (a+b)$. Do you see how a quantity that changes with the product of variables behaves differently than a quantity that changes with the sum of those variables?

yes

 

[mfb]

Or, try to find an example of two different shapes that have the same area and the same perimeter.

none

There are examples, but they are a bit harder to find. A triangle and a suitable trapezoid are the easiest examples.

 

[SSG-E]

t

There are examples, but they are a bit harder to find. A triangle and a suitable trapezoid are the easiest examples.

triangles in some cases?

 

[etotheipi]

Or, try to find an example of two different shapes that have the same area and the same perimeter.

Another exercise would be to try and prove which shape maximises $\frac{\text{Area}}{\text{Perimeter}}$. And what about $\frac{\text{Volume}}{\text{Surface Area}}$ in 3D?

 

[SSG-E]

Another exercise would be to try and prove which shape maximises $\frac{\text{Area}}{\text{Perimeter}}$. And what about $\frac{\text{Volume}}{\text{Surface Area}}$ in 3D?

Circle maximizes Area/perimeter. Sphere has the largest Volume/area.

 

[etotheipi]

Circle maximizes Area/perimeter. Sphere has the largest Volume/area.

It's right, but can you prove it?

 

[SSG-E]

It's right, but can you prove it?

In case of circle, it has infinite number of sides. In case of a sphere, For example, balloons are spherical, and they will assume the shape of minimum area

 

[etotheipi]

For example, balloons are spherical, and they will assume the shape of minimum area

You can come up with some physical arguments which might show that the radius of curvature of a small region of the surface of an elastic balloon filled with a gas at a uniform pressure must be constant everywhere, implying a sphere.

For a more mathematical approach, you might look at something like this.

 

[Svein]

And what about the ratio area/perimeter for this shape?

This is an example of a space-filling curve, a Sierpinski curve. The curve has a recursive definition, given one instance you can easily create the "next" and more complicated curve.

 

[WWGD]

Maybe a high-powered answer is that neither area nor perimeter are topological properties, so are not preserved under deformations. Same would go the other way around, if you fixed a perimeter you can obtain figures of different areas. Maybe you can do something fancier and describe in a general way the relation between area and perimeter.

 

[mfb]

There are examples, but they are a bit harder to find. A triangle and a suitable trapezoid are the easiest examples.

Thinking about it, two suitable trapezoids are a much easier example of two different shapes with the same area and perimeter.

 

[WWGD]

It may be an issue of using isometries which I assume would be the operations that would preserve these. Topology vs Geometry.

 

[jbriggs444]

Thinking about it, two suitable trapezoids are a much easier example of two different shapes with the same area and perimeter.

I have in mind flopping between trapezoid to a parallelogram.

 

[Infrared]

For a more mathematical approach, you might look at something like this.

And forgive the advertisement, but a proof for smooth curves was outlined in problem 3 of the February challenge thread (https://www.physicsforums.com/threads/math-challenge-february-2020.983823/)

 

[sysprog]

Please consider the following diagram:

The light blue shaded area is less than that area plus the grey shaded area (i.e. the area of the whole ABC triangle), but because 'the shortest distance between two points is a straight line', the DF line segment is shorter than the DE plus EF segments, which makes the perimeter path of the light blue shaded area, ABCFEDA, longer than that of the outside triangle. ABCA. or ABCFDA, despite the smaller area of the region enclosed by the longer path.