Expert Help with High School Geometry

How many sides does a circle have?

Get angle of point in circle with center (0, 0).
I have a circle with a centre at position (0, 0) and I have a point in the circle at (-5, 2). How can I get the angle of the point towards the centre?

Some textbooks define a trapezoid as a quadrilateral with exactly one pair of parallel sides, while other textbooks say that a trapezoid has at least one pair of parallel sides.
a. What are the consequences of choosing one of these definitions over the other? Which is correct?
b. Is a parallelogram a trapezoid? Why or why not?

How to calculate the volume of an ellipsoid with triple integral
I'm having some troubles since this morning on an exercise. I need to find the volume of the ellipsoid defined by $\frac{x^2}{a^2}+\frac{y^2}{a^2}+\frac{z^2}{a^2}\le <!--\; \le \; -->1$. So at the beginning I wrote $\{\begin{array}{c}-<!--\; -\; -->a\le <!--\; \le \; -->x\le <!--\; \le \; -->a\\ -<!--\; -\; -->b\le <!--\; \le \; -->y\le <!--\; \le \; -->b\\ -<!--\; -\; -->c\le <!--\; \le \; -->z\le <!--\; \le \; -->c\end{array}$
Then I wrote this as integral: ${\int <!--\; \int \; -->}_{-<!--\; -\; -->c}^c{\int <!--\; \int \; -->}_{-<!--\; -\; -->b}^b{\int <!--\; \int \; -->}_{-<!--\; -\; -->a}^{a}1dxdydz$.
I found as a result : 8abc
But I knew it was incorrect. I browsed this website and many others for 2 hours and yeah I found that we need to express b and c in terms of x. But I don't know why.. Is there an intuitive way to understand that ? I have already done some double integral with b expressed in terms of a but I never figured out why..

Distance between (-9,2) and (12,-8)?

Finding volume of pyramid in ${\mathbb{R}}^3$.
Assume we have a pyramid (triangular base) sitting in ${\mathbb{R}}^3$, i.e. we have the coordinates for the vertices A,B,C and D. How can we find the volume of the pyramid without using either its height or its base area and no advanced tools like integration?

Find the volume bounded by the following surfaces
I need to find the volume bounded by the following surfaces:
$x^2+y^2=2z$
$x^2+y^2=3-<!--\; -\; -->z$
I don't know how to proceed in solving this exercise, all I could think of was trying to make a system of these 2 and get $z=1$. I just wanna understand how am I supposed to proceed in finding the boundaries, no need to evaluate the integral.

$X,Y\sim <!--\; \sim \; -->U[0,1]$, what is the probability that $t^2+Xt+Y=0$ has a real root.
Let the continuous random variables X,Y be independent of each other and uniformly distributed on [0,1] i.e. $X,Y\sim <!--\; \sim \; -->U[0,1]$, what is the probability that $t^2+Xt+Y=0$ has a real root?
I am trying to solve it with Geometric probability models, knowing $X^2-<!--\; -\; -->4Y\ge <!--\; \ge \; -->0$. But I think the real answer should be related with r.v. and the properties of uniform distribution. What is the right way to solve the problem?

Probability of geometric mean being higher than 1/6
If we choose 2 numbers from [0,2], independent from each other,what is the probability of their geometric mean being higher than 1/6?

Why is $P(X>r)=q^r$?
If X follows a geometric distribution, where $p=$ probability of success and $q=$ probability of failure, why is $P(X>r)=q^r?$

Conditional expectation of a uniform distribution given a geometric distribution
Let N follow a geometric distribution with probability p. After the success of the experiment we define X, a uniform distribution from 1 to N. Both distributions are discrete. Find E[X|N].

Intuitive error in finding Volume of sphere using single integration.
To calculate the volume of a sphere of radius R, I considered a thin disc of radius $r=\mathbf{R}sin\mathrm{\theta <!--\; \theta \; -->}$, where $\mathrm{\theta <!--\; \theta \; -->}$ is the angle radius vector on the circumference makes with the axis of the disc. However I considered the volume $dV=$ $\mathrm{\pi <!--\; \pi \; -->}{r}^2\mathbf{R}d\mathrm{\theta <!--\; \theta \; -->}$, $\mathbf{R}d\mathrm{\theta <!--\; \theta \; -->}$ being the infinitesimal thickness of the disc, which is erroneous. The volume of the sphere comes out to be $\frac{3{\mathrm{\pi <!--\; \pi \; -->}}^2{\mathbf{R}}^3}{2}$. The correct thickness is $\mathbf{R}sin\mathrm{\theta <!--\; \theta \; -->}d\mathrm{\theta <!--\; \theta \; -->}$ which is found usually in other derivations gives the correct volume. However I do not understand what exactly is the issue in my approach and why the component of the curve length is considered.

Prove if $\{a_n\}$ $\to <!--\; \to \; -->$ $\mathrm{\infty <!--\; \infty \; -->}$, then $\{a_n\}$ is not bounded above. Give an indirect proof.

Distance between (9,2,0) and (0,6,0)?

X be a geometric random variable, show that $P[X\ge <!--\; \ge \; -->n]=(1-<!--\; -\; -->p{)}^{n-<!--\; -\; -->1}$

Geometric probability expressed as a geometric series
Suppose that we want to show that $P(X\le <!--\; \le \; -->n)=(1-<!--\; -\; -->p{)}^n$
Where X is a geometric random variable with probability of success p.
$P(X>n)$ can be written as a geometric series as follows:
${\mathrm{\Sigma <!--\; \Sigma \; -->}}_{X=n+1}^{\mathrm{\infty <!--\; \infty \; -->}}(1-<!--\; -\; -->p{)}^{n-<!--\; -\; -->1}=(1-<!--\; -\; -->p{)}^n$
But how do I proof this? What is a and r in this particular geometric series?

Unbiased sufficient statistic for 1/p of geometric distribution
Suppose that a random variable X has the geometric distribution with unknown parameter p, where the geometric probability mass function is:
$f(x;p)=p(1-<!--\; -\; -->p{)}^x,x=0,1,2,\dots <!--\; \dots \; -->;0<p<1$.
Find a sufficient statistic T(X) that will be an unbiased estimator of 1/p.
Now I know the population mean is $\frac{1-<!--\; -\; -->p}{p}$ and the sufficient statistic for p is a function of $\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{X}_i$. But I am unsure on how to proceed any help greatly appreciated!

Can we prove $\mathrm{\neg <!--\; \neg \; -->}R(x)⟹<!--\; ⟹\; -->\mathrm{\neg <!--\; \neg \; -->}R(y)$ by proving $\mathrm{\neg <!--\; \neg \; -->}\mathrm{\neg <!--\; \neg \; -->}R(y)⟹<!--\; ⟹\; -->\mathrm{\neg <!--\; \neg \; -->}\mathrm{\neg <!--\; \neg \; -->}R(x)$?
Can we also prove that by proving $R(y)⟹<!--\; ⟹\; -->R(x)$?

Probability of $-<!--\; -\; -->\frac{1}{4}\le <!--\; \le \; -->\sin <!--\; \; -->(ax)\le <!--\; \le \; -->\frac{1}{2}$?
We know that probability of having $\sin <!--\; \; -->(ax)>0$ for a random x is $\frac{1}{2}$.
Can we say something about the probability of the following condition?
$-<!--\; -\; -->\frac{1}{4}\le <!--\; \le \; -->\sin <!--\; \; -->(ax)\le <!--\; \le \; -->\frac{1}{2}$
Here, a and x are continuous and $x>0$ and $a>0$.

Trouble finding the volume using the shell method
A region is bounded by the line $y=3x+4$ and the parabola $y=x^2$ and is rotated about the line $x=4$.
First I found the limits of integration by finding the points of intersection. They are (-1,1) and (4,16). I then found the radius from the axis of rotation to be $r=4-<!--\; -\; -->x$ and the height to be $h=3x+4-<!--\; -\; -->x^2$.
I used the Volume formula ${\int <!--\; \int \; -->}_a^{b}2\mathrm{\pi <!--\; \pi \; -->}(radius)(height)dx$.
$V={\int <!--\; \int \; -->}_{-<!--\; -\; -->1}^{4}2\mathrm{\pi <!--\; \pi \; -->}(4-<!--\; -\; -->x)(3x+4-<!--\; -\; -->x^2)dx$
$2\mathrm{\pi <!--\; \pi \; -->}{\int <!--\; \int \; -->}_{-<!--\; -\; -->1}^4(x^3-<!--\; -\; -->7x^2+8x+16)dx$
Evaluating this integral I get the volume to be $280\mathrm{\pi <!--\; \pi \; -->}-<!--\; -\; -->{\displaystyle \frac{1055}{6}}$ but this is no where close to the actual volume. I believe that both my radius and height are right and that the formula is right. I've redid the problem a couple of times and ended up with the same answer, what am I missing here?