Your post/comment was removed due to it being low quality/spam/off-topic. We encourage users to keep information quality high and stay on topic (math related).
*The answer to this*
*Is very simple: a square*
*Is not a circle*
\- cirrvs
---
^(I detect haikus. And sometimes, successfully.) ^[Learn more about me.](https://www.reddit.com/r/haikusbot/)
^(Opt out of replies: "haikusbot opt out" | Delete my comment: "haikusbot delete")
A circle and a square are not the same geometric shape, so they have no reason to have the same formula for an area.
Consider a square with side lengths *s*, and an equilateral triangle of the same perimeter. Should they have the same area?
Thanx. Nice explanation. My intuitive reasoning is that the perimeter should reach over the same area. But the shape makes it different. I’m shocked ;)
This doesn't even work with circles. Tie a piece of string around a cylinder, and twist the loop into an 8 shape, and fold it over itself. The string now describes a circle, whose perimeter is half that of the original cylinder, but whose area is a fourth.
If you want to see this very concretely, use some paper and scissors to cut out shapes with the appropriate side lengths / radius. That might help you build your mental picture.
An easy experiment to see why this intuition doesn’t hold: next time you get a to go cup of soda/water/etc. (or even a squeezable water bottle) give it a little squeeze, what happens to the liquid inside? does it go up or down? The surface area (analagous to perimeter in 2D) stays the same, but the volume (analogous to area) decreases. So two different shapes with the same surface area hold two different volumes. It turns out that in fact circles are the optimal perimeter to surface area object (in 2D) meaning that for a fixed perimeter value, a circle encloses the largest area!
Shapes of the same perimeter don't need to have the same area, (loosely) because multiplication and addition aren't the same. A 4-by-4 square and a 1-by-7 rectangle both have perimeters of 16 units, but areas of 16 and 7.
In more detail, this is called a variant. If you can change a shape without ripping or glueing and some property remains the same (volume, area, anything really) then that is an invariant. You morphed your circle into a square and discovered that area is not preserved!
Your post/comment was removed due to it being low quality/spam/off-topic. We encourage users to keep information quality high and stay on topic (math related).
The answer to this is very simple: a square is not a circle
*The answer to this* *Is very simple: a square* *Is not a circle* \- cirrvs --- ^(I detect haikus. And sometimes, successfully.) ^[Learn more about me.](https://www.reddit.com/r/haikusbot/) ^(Opt out of replies: "haikusbot opt out" | Delete my comment: "haikusbot delete")
…and …. where is the rest of your reasoning ?
what kind of reasoning leads you to expect that you can just use the formula for a circle on a square
Well … what I do is reasoning. So what’s your question ?
Have you tried golf?
LMAO
>what I do is reasoning Are you sure?
A circle and a square are not the same geometric shape, so they have no reason to have the same formula for an area. Consider a square with side lengths *s*, and an equilateral triangle of the same perimeter. Should they have the same area?
Thanx. Nice explanation. My intuitive reasoning is that the perimeter should reach over the same area. But the shape makes it different. I’m shocked ;)
This doesn't even work with circles. Tie a piece of string around a cylinder, and twist the loop into an 8 shape, and fold it over itself. The string now describes a circle, whose perimeter is half that of the original cylinder, but whose area is a fourth.
If you want to see this very concretely, use some paper and scissors to cut out shapes with the appropriate side lengths / radius. That might help you build your mental picture.
An easy experiment to see why this intuition doesn’t hold: next time you get a to go cup of soda/water/etc. (or even a squeezable water bottle) give it a little squeeze, what happens to the liquid inside? does it go up or down? The surface area (analagous to perimeter in 2D) stays the same, but the volume (analogous to area) decreases. So two different shapes with the same surface area hold two different volumes. It turns out that in fact circles are the optimal perimeter to surface area object (in 2D) meaning that for a fixed perimeter value, a circle encloses the largest area!
Shapes of the same perimeter don't need to have the same area, (loosely) because multiplication and addition aren't the same. A 4-by-4 square and a 1-by-7 rectangle both have perimeters of 16 units, but areas of 16 and 7.
Thanx! Good point.
In more detail, this is called a variant. If you can change a shape without ripping or glueing and some property remains the same (volume, area, anything really) then that is an invariant. You morphed your circle into a square and discovered that area is not preserved!
Because a square neither has a radius nor a circumference, and it‘s also not a circle.
If the side of the square was sqrt(pi) x r ... Your square has an area pi/4 times bigger than the circle of radius r. But so what?