All Questions
214
questions
127
votes
8
answers
85k
views
Why is the derivative of a circle's area its perimeter (and similarly for spheres)?
When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle.
Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is ...
- 9,804
48
votes
16
answers
167k
views
Calculus proof for the area of a circle
I was looking for proofs using Calculus for the area of a circle and come across this one
$$\int 2 \pi r \, dr = 2\pi \frac {r^2}{2} = \pi r^2$$
and it struck me as being particularly easy. The only ...
- 1,140
35
votes
3
answers
5k
views
Why does area differentiate to perimeter for circles and not for squares?
I read this question the other day and it got me thinking: the area of a circle is $\pi r^2$, which differentiates to $2 \pi r$, which is just the perimeter of the circle.
Why doesn't the same ...
- 3,565
31
votes
6
answers
7k
views
Trying to understand why circle area is not $2 \pi r^2$
I understand the reasoning behind $\pi r^2$ for a circle area however I'd like to know what is wrong with the reasoning below:
The area of a square is like a line, the height (one dimension, length) ...
- 413
26
votes
3
answers
2k
views
Is it possible to turn this geometric demonstration of the area of a circle into a rigorous proof?
In this New York Times article, Steven Strogatz offers the following argument for why the area of a circle is $\pi r^2$. Suppose you divide the circle into an even number of pizza slices of equal arc ...
- 20.6k
16
votes
6
answers
3k
views
Why is the area of the circle $πr^2$? [duplicate]
I searched many times about the cause of the circle area formula but I did not know anything so ...
Why is the area of the circle $\pi r^2$?
Thanks for all here.
- 514
14
votes
6
answers
6k
views
How is the area of a circle calculated using basic mathematics?
Area of a circle is addition of circumference of layers of a onion. If n is radius of a onion then area is
$$
A = 2 \pi \cdot 1 + 2 \pi \cdot 2 + 2\pi \cdot 3 + \ldots + 2 \pi \cdot n
$$
which
$$
=...
- 301
9
votes
1
answer
150
views
Why does a circle appear when we square a polynomial whose inflection points are all on the $x$-axis?
I challenged myself to find a general formula for an $n$-degree polynomial with $n-2$ inflection points, all on the $x$-axis. Here is what I came up with (explanation is at the end).
$$\text{Even }n:...
- 25.6k
9
votes
1
answer
272
views
Simple proof of area of "rectangled" circle
Here is a simple problem which I would occasionally assign to my precalculus students and to my calculus students. The precalculus students always found a simpler answer. Sometimes it is possible to ...
- 21.9k
8
votes
1
answer
3k
views
How to turn this sum into an integral?
I have been trying to find the closed form of this sum to no avail. It was suggested to me to try and turn this sum into an integral and solve it like that. However, I am confused as to how to do ...
- 25.4k
8
votes
3
answers
262
views
Is there a simple formula for this simple question about a circle?
What is the average distance of the points within a circle of radius $r$ from a point a distance $d$ from the centre of the circle (with $d>r$, though a general solution without this constraint ...
- 520
7
votes
9
answers
25k
views
Calculate $\pi$ precisely using integrals?
This is probably a very stupid question, but I just learned about integrals so I was wondering what happens if we calculate the integral of $\sqrt{1 - x^2}$ from $-1$ to $1$.
We would get the surface ...
- 1,283
7
votes
3
answers
13k
views
Is the tangent function (like in trig) and tangent lines the same?
So, a 45 degree angle in the unit circle has a tan value of 1. Does that mean the slope of a tangent line from that point is also 1? Or is something different entirely?
- 81
7
votes
1
answer
176
views
A chain of circles of radius $1/n^p$ is tangent to the $x$-axis. What is the horizontal length of the chain?
I recently discovered that, if a chain of circles of radius $1/n^2$, where $n\in\mathbb{N}$, is tangent to the $x$-axis, then the the horizontal length of the chain is exactly $2$.
This can be shown ...
- 25.6k
6
votes
3
answers
849
views
The area of circle
The question is to prove that area of a circle with radius $r$ is $\pi r^2$ using integral. I tried to write $$A=\int\limits_{-r}^{r}2\sqrt{r^2-x^2}\ dx$$
but I don't know what to do next.
user164524