Math Is Fun Forum

paulb203

Member

Registered: 2023-02-24

Posts: 406

Why πr^2?

I know there are videos etc explaining why but I thought I would try to find a way to understand this myself.

Imagine a version of πr^2, but instead of being for the area of a circle, it’s for the area of a square.
Call it sqi ar ^2
sqi = the ratio of the ‘diameter’ of the square to the square’s ‘circumference’ (the ‘diameter’ of the square being a vertical or horizontal line through the square’s centre, and the ‘circumference’ being its perimeter).
So, sqi=4 (like π=3.14...)
ar = the ‘radius’ of the square (half it’s ‘diameter’, just like with a circe and it’s radius)

Now take a square 4 cm by 4cm
And apply sqi ar ^2
Which gives us 4x2^2
Which = 16cm^2
Which matches with the 16cm^2 we would get from the conventional way of finding the area of the square.

This helps make sense of πr^2, for me at least
Any thoughts?

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"The secret of getting ahead is getting started."
Mark Twain

KerimF

Member

From: Aleppo-Syria

Registered: 2018-08-10

Posts: 282

Re: Why πr^2?

You did well. It is always good to see something from different angles.

For instance, since I have no more a person to tell me if my solution of a problem is wrong or right, I used to find out different ways (two in the least) to solve it.

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Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).
But only a human may have the freedom and ability to oppose his natural robotic nature.
But, by opposing it, such a human becomes no more of this world.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,759

Re: Why πr^2?

hi paulb203

It looked to me like this should be 'provable' using algebra but I've come unstuck with it.  I don't think I'm properly following what you're suggesting.

If a square has side 'a' then the 'diameter' = a and so the 'radius' = a/2 and the perimeter = 4a

sqi = the ratio of the ‘diameter’ of the square to the square’s ‘circumference’

So the ratio of diameter/circumference = a/4a = 1/4.  Ah! Think I've just spotted what to do.

Ratio of circumference/diameter = 4

Then 'area' = 4 x (a/2)^2 = 4 (a^2)/4 = a^2

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

paulb203

Member

Registered: 2023-02-24

Posts: 406

Re: Why πr^2?

You did well. It is always good to see something from different angles.

For instance, since I have no more a person to tell me if my solution of a problem is wrong or right, I used to find out different ways (two in the least) to solve it.

Thanks, Kerim F

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"The secret of getting ahead is getting started."
Mark Twain

paulb203

Member

Registered: 2023-02-24

Posts: 406

Re: Why πr^2?

hi paulb203

It looked to me like this should be 'provable' using algebra but I've come unstuck with it.  I don't think I'm properly following what you're suggesting.

If a square has side 'a' then the 'diameter' = a and so the 'radius' = a/2 and the perimeter = 4a

sqi = the ratio of the ‘diameter’ of the square to the square’s ‘circumference’

So the ratio of diameter/circumference = a/4a = 1/4.  Ah! Think I've just spotted what to do.

Ratio of circumference/diameter = 4

Then 'area' = 4 x (a/2)^2 = 4 (a^2)/4 = a^2

Bob

Thanks, Bob.

And yes, to all of that.

Just checking I've followed your algebra.

Area = 4 x (a/2)^2 = 4 (a^2)/4 = a^2

First part; 4x(a/2)^2 = 4(a^2)/4
So, with (a^2)^2 you’ve squared the a, and squared the 2?

Second part; 4 (a^2)/4 = a^2
You have 4 times, a^2 over 4, and you’ve cancelled the two 4s, yeah?

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"The secret of getting ahead is getting started."
Mark Twain

Bob

Administrator

Registered: 2010-06-20

Posts: 10,759

Re: Why πr^2?

Correct!

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

paulb203

Member

Registered: 2023-02-24

Posts: 406

Re: Why πr^2?

Thanks, Bob.

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"The secret of getting ahead is getting started."
Mark Twain