Math Is Fun Forum
You are not logged in.
- Topics: Active | Unanswered
#1 2024-08-05 12:31:30
- Frank#smith[Kfyzeiuqujbpy
- Guest
Why πr2?
Why is the formula for the area of a circle πr2 and not something else? Is there a deeper mathematical reason behind this seemingly arbitrary formula, or is it just a result of the way we've defined circles and areas?
#2 2024-08-05 13:02:44
- Jai Ganesh
- Administrator
- Registered: 2005-06-28
- Posts: 48,395
Re: Why πr2?
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
Offline
#3 2024-08-05 15:32:21
- Phrzby Phil
- Member
- From: Richmond, VA
- Registered: 2022-03-29
- Posts: 50
Re: Why πr2?
The formula for the area of a circle is derived from the integral calculus, which deals with smoothly changing shapes.
Hard to explain otherwise.
World Peace Thru Frisbee
Offline
#4 2024-08-05 19:41:27
- Bob
- Administrator
- Registered: 2010-06-20
- Posts: 10,621
Re: Why πr2?
hi Frank smith
Welcome to the forum.
It's a good question.
If you look here: https://www.mathsisfun.com/geometry/cir … ctors.html
you'll see a way to establish the formula by dividing a circle into sectors and letting the number of sectors become infinite.
But there's still a question because this method uses the formula for the circumference, so is there a proof for this too?
I cannot find one on the MIF site; I'm sure you could 'google' it.
I've got a method that works out the perimeter of a regular polygon and lets the number of sides tend to infinity so the polygon becomes a circle.
It uses sin(x) tends to x as x tends to zero (x in radians). There a justification* for that here:
https://www.mathsisfun.com/geometry/radians.html
* The method I learnt at school uses the formula for circumference so it isn't suitable here as it creates a circular argument (no pun intended).
Bob
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 
Offline