Simple questions on Maths

Lee012lee SG

Why is the area of a circle equal to pi x radius^2 ?

How is the value of pi determined ?

The answers seem cannot be found in the primary maths textbooks leh ?

lianamaster SG

I dun derive it...I use it.

105090 SG

IMO, the value of pi happens to be a constant

FirePig SG

One way of visualising is to cut the circles into many equal pies. When you cut into many many equal and small pies (size of a hair with a rounded top) each of radius r.

Then you arrange the pies to form a rectangle and eventually you will get roughly a rectangle of breadth r and length pi x r.

In fact, you will realise the area of circle is indeed length x breadth = (pi x r) x r.

FirePig SG

Here la

animation here

http://www.wku.edu/~tom.richmond/Pir2.html

Lee012lee SG

IMO, the value of pi happens to be a constant

Then, why is pi a constant and why pi is usually taken as 3.14 or 22/7 leh ie how is the value of pi determined ?

Lee012lee SG

Here la

animation here

http://www.wku.edu/~tom.richmond/Pir2.html

Thanks for the answer. But how come our primary maths textbooks do not explain why is the area of a circle equal to pi x radius^2 ? Sigh.

In the link, it has taken the base to be pi x radius but the length in the diagram is not a straight line but makes up of many arcs.

so, does this mean that the formula that we have been using to calculate the area of a circle is inaccurate ??????

eagle SG

Thanks for the answer. But how come our primary maths textbooks do not explain why is the area of a circle equal to pi x radius^2 ? Sigh.

I learned in pri sch that it was originally found from observations by mathematicians that the area of circles are proportional to their respective radii squared.

From there, they found pi.

Pi is usually taken as 3.14 or 22/7 as a rough estimate. The actual value is 3.141592356589............ => It's an irrational number.

Certain series have been derived to find the value of pi.

Lee012lee SG

I learned in pri sch that it was originally found from observations by mathematicians that the area of circles are perpendicular to their respective radii.

From there, they found pi.

Pi is usually taken as 3.14 or 22/7 as a rough estimate. The actual value is 3.141592356589............ => It's an irrational number.

Certain series have been derived to find the value of pi.

I do not understand why the area of a circle is perpendicular to its radius ie area is a region and radius is a line. So, how to image that they are perpendicular to one another and how from there, the pi is found. Explain a bit more leh.

From the link provided by Firepig, pi is the ratio of circumference to two radius or diameter. But the link has not explain how the value of pi is determined ie what series is used to find the value of pi as explained by Eagle. Explain a bit more leh which series is used and why the series can be used to find the value of pi leh.

I understood the link provided by Firepig on how the area of a circle is derived but
in the link, it has taken the base to be pi x radius but the length in the diagram is not a straight line but makes up of many arcs.

so, does this mean that the formula that we have been using to calculate the area of a circle is inaccurate ??????

eagle SG

so, does this mean that the formula that we have been using to calculate the area of a circle is inaccurate ??????

used wrong word

I meant proportional... Too tired...

eagle SG

I think this is a good read

http://en.wikipedia.org/wiki/Pi

$\pi = \frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}\cdots\!$

Lee012lee SG

Some parts in the link are difficult to understand but the link is excellent in that it really provides a detailed explanation of how the value of pi is determined lor.

Thanks. D.

Lee012lee SG

Since the value of pi cannot be determined exactly, so the formula of the area of a circel pi x radius^2 cannot be determined exactly too. So, the pi x radius^2 formula will only give us approximate area of the circle. D.

eagle SG

yep

but if you think carefully, this way, nothing in this world can be measured exactly as well. Well... almost...

SlyS SG

When a circle's diameter is 1, its circumference is π. π is a trancendental number, which means that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) could ever produce it.

Referred to Wikipedia =)

UltimaOnline SG

(Eagle beat me to posting the Wikipedia entry...)