Area of a circle and pi and generally area

[RadiantL]

So I always wondered why you multiply by pi when you're finding an area of a circle, for a rectangle you multiply by length and width, I guess that makes sense...

How I see multiplying a length and width is if you have a length of 5 cm and a width of 4 cm, I imagine you just stack 4, 5 cm sticks on top of each other and you get the area is this how it works?

Anyway I have trouble memorizing formulas for area of a circles and cylinders and such and I am sure I would be better at recalling them if I knew why they are... what they are.

A = (pi)(r)^2

What's with pi?

 

[fashizzle]

Also interested in this as well, I've always thought that circumference of a full circle arises from 2*pi*r because radius multiplied by central angle gives you the arc length, and by taking the integral of this with respect to r, you get pi*r^2.

and if you integrate a circle of radius R using polar coordinates, you can change the value of theta to give you the area of swathes within the circle, so pi will change on the basis of whether or not you're dealing with a full circle.

 

[gsal]

Here is a whole and very nice explanation of why the area of a circle is what it is:

http://www.worsleyschool.net/science/files/circle/area.html

 

[RadiantL]

Here is a whole and very nice explanation of why the area of a circle is what it is:

http://www.worsleyschool.net/science/files/circle/area.html

Wow, that's brilliant haha thanks so much!

 

[ArcanaNoir]

Very nice website. I loved the area explanation.

 

[all-black]

Here is a whole and very nice explanation of why the area of a circle is what it is:

http://www.worsleyschool.net/science/files/circle/area.html

nice explanation..

 

[dextercioby]

I was famous for saying this here on PF years ago, so I say it again:

Strictly speaking, the circle has no area. If it did, it would be 0.

 

[Deveno]

I was famous for saying this here on PF years ago, so I say it again:

Strictly speaking, the circle has no area. If it did, it would be 0.

well, ok. we should be saying "the area enclosed by a circle", or the area of a circular region (or the area of a regular disc). do you want fries with that?

 

[HallsofIvy]

Or "area of the disk".

 

[Pengwuino]

Here is a whole and very nice explanation of why the area of a circle is what it is:

http://www.worsleyschool.net/science/files/circle/area.html

But why is the circumference of a circle $2 \pi r$

 

[I like Serena]

It's because pie is defined as the ratio of your upper lip and the length of the curve around your mouth and eyes.

 

[Deveno]

But why is the circumference of a circle $2 \pi r$

i honestly don't know why, i didn't design this universe.

 

[Redbelly98]

But why is the circumference of a circle $2 \pi r$

I realize you're speaking tongue-in-cheek, but the real question would be "Why is the circumference of a circle proportional to its diameter?" π is simply the proportionality constant for that relation, by definition.

 

[Pengwuino]

I realize you're speaking tongue-in-cheek, but the real question would be "Why is the circumference of a circle proportional to its diameter?" π is simply the proportionality constant for that relation, by definition.

Because the units work out.

I love physics.

 

[DivisionByZro]

Because the units work out.

I love physics.

This made me cringe.
To each their own I suppose.

 

[sankalpmittal]

But why is the circumference of a circle $2 \pi r$

Pi is defined as the ratio of the circumference of the circle to the diameter of that circle originally.

Do this experiment : Draw circles with compass of different radius and measure their circumference by using a thread or a string : Enclose circle boundary with thread and then measure that part of thread with a ruler.

You will find that : C₁/D₁=C₂/D=...=C_n/D_n = pi

So C/D = pi
or C=2*pi*R since D=2R

Here are proofs of area of circle : https://www.physicsforums.com/showthread.php?t=529014

gsal , that site is also nice which you gave !

 

[Deveno]

Pi is defined as the ratio of the circumference of the circle to the diameter of that circle originally.

i have a problem with this, which is:

what is "circumference"? a closely related question is: "what is diameter"?

if you answer something like: "diameter is the length of the longest possible line segment across the circle", or perhaps "the length of any line segment from a point on the circle, through the center of a circle, and terminating at a point on the opposite side of the circle", i would still want to know, what is this "length" thing we are talking about. how do we tell when two lengths are the same, and how do we tell which of two unequal lengths is longer?

if you answer, "we measure them, and compare", i again ask, "how is it we measure things"?

what KINDS of objects qualify as "measurements of lengths", and how do we know that this is a "proper" description (logically consistent)?

now, this is kind of unfair, i actually know the answers to these questions. but i would humbly submit, that when a 6-th grader, for example, is given the definition:

"pi is the ratio of a circle's circumference to it's diameter"

there are several "hidden assumptions"

1) circumference can be unambiguously measured
2) diameter can be unambiguously measured
3) circumference and diameter are "comparable" (the same kind of number), and we may form their ratio (suggests a notion of division)
4) this ratio is always the same, regardless of the length of the diameter (pi is constant)

all of these statements are provable, but some of them are subtler than others.

 

[logmode]

You have me thinking now, and I hope it’s OK to ask a question here. I am not sure, but is this true 2 ∏ = circumference?

 

[BloodyFrozen]

You have me thinking now, and I hope it’s OK to ask a question here. I am not sure, but is this true 2 ∏ = circumference?

$$C=2\pi r ~~ or ~~C=\pi D$$

 

[logmode]

Am I correct, ∏ is 3.14 radians. If so, 360 degrees is a circle, which is the circumference. 360 degrees converted to radians is 360 x ∏/180 = 2∏, Where am I thinking wrong?

 

[checkitagain]

Am I correct, ∏ is 3.14 radians. If so, 360 degrees is a circle,
which is the circumference. 360 degrees converted to radians is 360 x ∏/180 = 2∏,
Where am I thinking wrong?

$\pi \ \ radians \ \ is \ \ approximately \ \ 3.14 \ \ radians.$

(You need the units on each number for equality.)


$360°\bigg(\dfrac{\pi \ radians}{180°}\bigg) \ = \ 2\pi \ radians$

 

[logmode]

Thank you, yes units are important, but what about the answer? C=2∏,
360* ( ∏ radians/180* ) = 2∏ radians = circumference in radians = 360* (.0174) = 6.28, so two PI equals circumference. C=2∏ ,but what was said was C=2∏r.
sorry for my confusion.

 

[logmode]

My mistakes were thinking that PI was only in radians, and 3.14 radians is equal to half the circumference. But I now know that PI is a little bigger (3.14 bigger) than three diameters of the circle, and radian is the length of radius. Thank you!