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 Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion

 March 22nd, 2018, 05:02 AM #1 Member   Joined: Oct 2017 From: Japan Posts: 62 Thanks: 3 Math for Fun, Area of a circle If anyone is interested I came across an old and simple method to get the area of a circle and made a lesson out of it. Do you know that you can easily get the formula of the area of a circle from very basic math, without using calculus? In this lesson I will show you how. Please subscribe if you like and let me know if you have any question. March 22nd, 2018, 08:45 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 Well, I think this is a matter of "sweeping the hard part under the rug". That is, it uses the fact that the circumference of a circle is $\displaystyle 2\pi r$ - and introducing $\displaystyle \pi$ itself is the hard part! Last edited by skipjack; March 22nd, 2018 at 10:46 AM. March 22nd, 2018, 10:58 AM #3 Senior Member   Joined: Apr 2014 From: Glasgow Posts: 2,161 Thanks: 734 Math Focus: Physics, mathematical modelling, numerical and computational solutions When I was doing my physics degree in 2002, I cam up with the following: Consider a regular polygon with side length $\displaystyle a$ and number of sides $\displaystyle n$. Then split up the regular polygon into triangles by drawing a line from each vertex to the centre of the shape. Each of those triangles can now be split into 2 right-angled triangles by drawing a bisecting line (of length $\displaystyle h$) from the centre of the shape to the middle of each side of the polygon. You then have the following: The angle, $\displaystyle \theta$, between two of those straight lines at the centre of the polygon is $\displaystyle \theta = \frac{2 \pi}{2n} = \frac{\pi}{n}$ but $\displaystyle \tan \theta = h / (a/2) = \frac{2h}{a}$ Therefore $\displaystyle h = \frac{a}{2} \tan \theta = \frac{a}{2} \tan \left(\frac{\pi}{n}\right)$ The area of the triangle is $\displaystyle A_{tri} = \frac{1}{2} base \times height = \frac{1}{2} \times \frac{a}{2} \times h = \frac{a^2}{8}\tan \left(\frac{\pi}{n}\right)$ Therefore, the area of the regular polygon is this area times the number of triangles, which is $\displaystyle 2n$, therefore $\displaystyle A = 2n A_{tri} = \frac{a^2n}{4}\tan \left(\frac{\pi}{n}\right)$ This is valid for $\displaystyle n \ge 3$. Let's check: $\displaystyle n = 3, A = \frac{\sqrt{3}}{4} a^2$ $\displaystyle n = 4, A = a^2$ etc... Alternatively, if we don't have $\displaystyle a$ but we do have $\displaystyle h$, we can resubstitute to get: $\displaystyle A = h^2n\cot \left(\frac{\pi}{n}\right)$ We can now consider the limit $\displaystyle n \rightarrow \infty$. With some nice maths, you can obtain $\displaystyle \pi r^2$ I thought it was original until I found a super old 1950s textbook with a full description there Thanks from jonah and Sebastian Garth March 22nd, 2018, 11:51 AM #4 Global Moderator   Joined: Dec 2006 Posts: 20,972 Thanks: 2222 What does "some nice maths" mean? Thanks from Benit13 March 23rd, 2018, 02:12 AM   #5
Senior Member

Joined: Apr 2014
From: Glasgow

Posts: 2,161
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Math Focus: Physics, mathematical modelling, numerical and computational solutions
Quote:
 Originally Posted by skipjack What does "some nice maths" mean?
L'Hôpital's rule and a couple of small angle approximations. I'll put it here if I get time. Last edited by skipjack; March 23rd, 2018 at 04:32 AM. March 23rd, 2018, 06:11 AM #6 Member   Joined: Oct 2017 From: Japan Posts: 62 Thanks: 3 I agree on that but still find the method interesting nonetheless. Tags area, circle, fun, math Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post ma78s Trigonometry 1 March 21st, 2017 12:55 AM skimanner Math 12 July 8th, 2016 07:55 AM yeoky Algebra 4 May 3rd, 2014 01:06 AM tiba Algebra 4 June 20th, 2012 11:45 AM gus Algebra 1 April 17th, 2011 04:25 PM

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