Finding Order Of The Area Of Geometric Plain Figures. [When perimeter is the same]

Hello maths bugs(🐞) & hivers(🐝)
I hope you are strong and stout and having a good time.

It's been about two week I have not posted anything.Actually I was quite busy in work and managing time to prepare figures I use in post was not possible. Sorry for that. I think from now I can be normal as I was previously.

You must have noticed today's cover.I have come up with Geometric figures and their relation of areas when the same perimeter of them is given.I have used three simple Geometric figure of same perimeter.For them to find out order of their areas is easy and many of already know it.

I will start proving taking easy cases and will go deeper to conclude general concept for that we can deal with any plain figure in this regard.

Before proceeding further we need to know about the basic concept of the figures I have shown in the above figure;specially about their perimeter and areas.So, let me introduced them first.

1️⃣Circle:

If radius(r) of a circle is given, the perimeter is written as 2πr unit and area is writen as πr² unit². Check it in the following drawn figure:

2️⃣Square:

If a side(a) of an square is given, the perimeter is written as 4a unit and the is written as a² unit².Check it in the following slide:

3️⃣Triangle:

For an easy calculation process,I will take an equilateral triangle.If a side(a) of a equilateral triangle given, the perimeter is written as 3a and it's area is know as √3a²/4 unit².Check it in the figure below:

👁️‍🗨️👁️‍🗨️When perimeter is the same:

Now it's time for comparing their area. let each of the above three figures have the same perimeter 3x unit. For the equilateral triangle the area will be √3a²/4 unit². For square the area is 9a²/16 unit² and that of the circle is 9a²/4π unit². Check details in the following figure:

Please calculate yourself that 9x²/4π is greater than 9x²/16. I was running short of space.😂😂

Finding relation of area for above three cases is an easy task to do as you have seen.Suppose you have two figures such as an square and a rhombus having same perimeter. Check details in the following frame:

So a square has greater area than that of a rhombus if the perimeter is same. If we take all of them together the descending order will be like this:

TRICKS TO LEARN IT:

When perimeter is the same:

1. The plain Geometric figure with more number of sides will have more area than a figure with a less number of sides.Consider a circle has infinite number of sides.

2. If there are many shapes having same number of sides, consider the more close to round shape will have greater area.

When area is the same:The the perimeter in this case will be in reverse order of the previous case. Check it below:

If there is any typing mistake specially in the frames, I am sorry for that. I tried by best. To prepare a frame is very time consuming. So editing again is not possible 😃.please just ignore it if you can.

I hope you liked explanation of today's topic.

Thank you so much for visiting and showing love for my hard work.

Have a good day.

All is well.

Regards: @meta007