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 ...
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 ...
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 ...
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) ...
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 ...
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.
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
$$
=...
9
votes
1
answer
150
views
Why does a circle appear when we square a polynomial whose inflection points are all on the $x$-axis?
I challenged myself to find a general formula for an $n$-degree polynomial with $n-2$ inflection points, all on the $x$-axis. Here is what I came up with (explanation is at the end).
$$\text{Even }n:...
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 ...
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 ...
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 ...
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 ...
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?
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 ...
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.
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 (...
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$
...
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$,...
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}{...
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 ...
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 ...
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 ...
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 ...
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 ...
5
votes
2
answers
359
views
Geometric Identities involving $π^2$
Are there any known geometric identities that have $π^2$ in the formula?
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 ...
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 ...
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 ...
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, ...
4
votes
1
answer
222
views
Why do derivatives of certain equations relating to circles yield other similar equations? [duplicate]
Possible Duplicate:
Why is the derivative of a circle's area its perimeter (and similarly for spheres)?
We all know that the volume of a sphere is:
$V = \frac{4}{3}\pi r^{3}$
and its ...
4
votes
1
answer
551
views
Is this a valid proof for the area of a circle?
My teacher challenged my class to prove that the area is
$$A=\pi r^2.$$
We recently learned about Riemann sums, so I thought it would be possible to apply them to them to deriving the formula for the ...
4
votes
2
answers
302
views
Area of $(x-3)^2+(y+2)^2<25: (x,y) \in L_1 \cap L_2$
Two lines $(L_1,L_2)$ intersects the circle $(x-3)^2+(y+2)^2=25$ at the points $(P,Q)$ and $(R,S)$ respectively. The midpoint of the line segment $PQ$ has $x$-coordinate $-\dfrac{3}{5}$, and the ...
4
votes
1
answer
2k
views
Circular Argument in Proof of Circumference of a Circle using Calculus
I have some doubts in this demonstration:
Prove the Circumference of a Circle is $C=2 \pi r$
The equation of a circle is
$(x-h)^2 + (y-k)^2 = r^2 $
To do computations easier let's consider a circle ...
4
votes
2
answers
5k
views
Equation of a tangent on a circle given the gradient and equation of the circle
My maths teacher told me this problem was impossible without knowledge of implicit differentiation: is she right?
You are given the equation of the circle $\left(x+2\right)^2+\left(y-2\right)^2=16$ , ...
3
votes
3
answers
376
views
The wrong way of finding the average distance between two points on a circle
I was trying to find the average distance between two points on a circle and got the following result.
Why is my method wrong?
3
votes
1
answer
428
views
circles tangent to exponential curve
Circle $C_1$ is tangent to the curves $y=e^x$ and $y=-e^x$ and the line $x=0$, and for $n>1$ circle $C_n$ is tangent to both curves and to $C_{n-1}$, how can I find the radius of any circle $C_k$?
...
3
votes
2
answers
186
views
Area of a circle which is interior to the parabola
I'm trying to solve the following problem:
Find the area of the circle $x^2+y^2=8$, which is interior to the
parabola $y^2=2x$.
I have my own solution and I want to verify whether it is correct. ...
3
votes
3
answers
2k
views
Prove that 4 points belong to the same circle by using complex numbers
I have Z1, Z2, Z3, Z4 and they are all complex numbers. I want to prove that they belong on the same circle(C) and its center is O where O = 3
How do I do that? (They actually have equations, I just ...
3
votes
3
answers
1k
views
Find the radius of the largest circle
In the accompanying diagram, a circle of radius $r$ is tangent to both sides of the right-angled corner. What is the radius of the largest circle that will fit in the same corner between the larger ...
3
votes
2
answers
175
views
Simplify a formula with 449 terms - Radical circle
Context
The other day I wanted to answer this question.
Which is now closed so doesn't accept answers (but this isn't the important part).
Since I didn't know the topic I went to look it up but I ...
3
votes
3
answers
635
views
Slope of the tangents to the circle $x^2+y^2-2x+4y-20=0$
Find the slope of the tangents to the circle
$x^2+y^2-2x+4y-20=0$.
After I arranged into a standard form, which is $(x-1)^2 +(y+2)^2=25$
Centre point is $(1,-2)$ radius is $5$ unit.
Do I need to do ...
3
votes
3
answers
151
views
Why does $\vec{F(t)} \cdot \vec{v(t)} = 0$ lead to a circular motion?
Here is a mathematical proof that any force $F(t)$, which affects a body, so that $\vec{F(t)} \cdot \vec{v(t)} = 0$, where $v(t)$ is its velocity cannot change the amount of this velocity.
Further, ...
3
votes
3
answers
8k
views
If $x^2 + y^2 + Ax + By + C = 0 $. Find the condition on $A, B$ and $C$ such that this represents the equation of a circle.
If $x^2 + y^2 + Ax + By + C = 0 $. Find the condition on $A, B$ and $C$ such that this represents the equation of a circle.
Also find the center and radius of the circle.
Here's my solution, I'm ...
3
votes
1
answer
2k
views
Define polar coordinates of circle at origin and circle with radius $R$.
Question:
(i) Define in polar coordinates $r = f(\alpha)$ the origin-centred circle with radius $R$. Specify the domain range for the polar coordinate $\alpha$.
(ii) Define in polar coordinates $r = ...
3
votes
3
answers
306
views
Finding the radii that maximizes and minimizes the area of four inscribed circles in an equilateral triangle.
An equilateral triangle with side length $1$ unit contains three identical circles $C_1$, $C_2$ and $C_3$ of radius $r_1$, each touching two sides of the triangle. A fourth circle $C4$ of radius $r_2$ ...
3
votes
1
answer
112
views
Ratio of Perimeter^3 to the Area of an Isoceles Triangle.
I am in trouble with the following question:
QUESTION
ABC is an isosceles triangle inscribed in a circle of radius $r$. If $AB=AC$ and $h$ is the altitude from $A$ to $BC$ and $p$ be the perimeter ...
3
votes
1
answer
270
views
What does Spivak want me to do?
This goes on in Chapter 8, on least upper bounds and related topics. I have proven $(a),(b),(c)$.
The sketch is.
$(a)$ If $\{a_n\}$ is a sequence of positive terms such that $$a_{n+1}\leq a_n/2$$ ...
3
votes
1
answer
197
views
Plotting a fix number of points evenly across surface area of a circle or sphere.
I would like to design an LED sphere, but I am having some trouble deciding on the placement of LEDs evenly across its surface area.
I would like there to be 32 evenly spaced LEDs across the ...
3
votes
0
answers
181
views
Question about the second derivative of a circle
I am trying to find the second derivative of a general circle but I can't seem to get the right answer.
My working goes as follows:
$$
(x-a)^2+(y-b)^2=R^2
$$
$$
2(x-a)+2(y-b)*\frac{dy}{dx}=0
$$
$$
\...