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 ...
- 9,805
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 ...
- 3,575
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 ...
- 20.7k
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:...
- 25.7k
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
177
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 ...
- 25.7k
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
6
votes
4
answers
2k
views
Definite integral: $\int^{4}_0 (16-x^2)^{\frac{3}{2}}\,dx$
The following integral can be computed using the substitution $x = 4\sin\theta~$ and then proceeding with $dx = 4\cos\theta~ d\theta~$, and evaluating the integral of $\cos^4\theta$ :
$$\int^{4}_0 (...
- 1,498
6
votes
5
answers
6k
views
Why is the distance between two circles/spheres that don't intersect minimised at points that are in the line formed by their centers?
From GRE 0568
From MathematicsGRE.Com:
I'm guessing the idea applies to circles also?
Is there a way to prove this besides the following non-elegant way?
Form a line between centers $C_1$ and $C_2$
...
- 13.7k
6
votes
3
answers
4k
views
Real numbers $x,y$ satisfies $x^2+y^2=1.$If the minimum and maximum value of the expression $z=\frac{4-y}{7-x}$ are $m$ and $M$
Real numbers $x,y$ satisfies $x^2+y^2=1.$If the minimum and maximum value of the expression $z=\frac{4-y}{7-x}$ are $m$ and $M$ respectively,then find $2M+6m.$
Let $x=\cos\theta$ and $y=\sin\theta$,...
- 5,868
6
votes
4
answers
218
views
Tangent to $x^2+y^2-6x-6y=-13$ and $x^2+y^2+2x+2y=-1$
Considering the circles $\lambda: x^2+y^2-6x-6y=-13$ and $\theta: x^2+y^2+2x+2y=-1$ find the line simultaneously tangent to them.
I found the implicit derivative of those two,
$\lambda: y'=-\frac{x-3}{...
- 109
6
votes
5
answers
728
views
The hardest geometry question with "a triangle" and "a circle" - Circle intersecting triangle equally in 5 parts
I received this question long time ago from one of my old friends who is mathematician/physicist. He called it the hardest geometry question with "a triangle" and "a circle". I am ...
- 725
6
votes
1
answer
3k
views
Is *njwildberger* wrong about area and circumference of a circle?
In this video, njwildberger says that the area and circumference of a circle are proof-less theorems. But I heard that we can derive both the area and circumference of a circle using calculus? So are ...
6
votes
2
answers
170
views
How do I evaluate the following limit?
I understand that I need to somehow use that $PR=AP=AQ$ as the point $A \to P$. But beyond that, I am unable to use that information to find $OB$.
This problem is from the textbook "Calculus with ...
- 63
6
votes
1
answer
235
views
Triangle and Circle maximization problem
So I was playing around GeoGebra and found this thing out, I don't know if this problem has a name or something.
Triangle ABC is inscribed inside a circle, from point D which is located inside the ...
- 61
6
votes
2
answers
2k
views
Shortest path between two points around an obstacle?
I'm trying to figure out a problem that goes like this:
A particle originally placed at the origin tries to reach the point $(12,16)$ whilst covering the shortest distance possible. But there is a ...
- 71
6
votes
0
answers
131
views
Is this a valid way of deriving the area of a circle?
On the Wikipedia article about deriving the area of a circle, it mentions that the formula
$$
\text{area} = \pi r^2
$$
can be derived by evaluating the integral
$$
2 \int_{-r}^{r} \sqrt{r^2-x^2} \, dx ...
- 20.7k
5
votes
2
answers
359
views
Geometric Identities involving $π^2$
Are there any known geometric identities that have $π^2$ in the formula?
- 1,844
5
votes
2
answers
23k
views
Rotate a point on a circle with known radius and position
Having a circle $\circ A(x_a, y_a)$ of radius $R$ and a point on the circle $B(x_b, y_b)$, how can we rotate the point with a known angle $\alpha$ (radians or degrees, it doesn't really matter) on the ...
- 567
5
votes
2
answers
193
views
Question from applications of derivatives.
Prove that the least perimeter of an isoceles triangle in which a circle of radius $r$ can be inscribed is $6r\sqrt3$.
I have seen answer online on two sites. One is on meritnation but the problem is ...
- 847
5
votes
1
answer
731
views
Average distance to a non-central point in a circle
If I pick a point within the circle about the origin of radius $R$, say $(r,\theta) = (0.5 R, \frac{\pi}{2})$, what is the average distance of all other points to that point?
Things which are ...
- 61
4
votes
3
answers
582
views
Help with Calculus Optimization Problem!
We wish to construct a rectangular auditorium with a stage shaped as a semicircle of radius $r$, as shown in the diagram below (white is the stage and green is the seating area). For safety reasons, ...
- 141