Expert Help with High School Geometry

The sum of the measures of the interior angles of a polygon is ${720}^{\circ <!--\; \circ \; -->}$ . What type of polygon is it?

Check my work? Finding line that divides rotational volume into equal parts. Not getting right answer.
I need to rotate the area between the curve $y=x^2$ and $y=9$, bounded in the first quadrant, around the vertical line $x=3$. I then must find the height (m) of the horizontal line that divides the resulting volume in half.
I've been trying to set up two integrals with washers. One from ${\int <!--\; \int \; -->}_m^9{3}^2-<!--\; -\; -->(3-<!--\; -\; -->x{)}^{2}dy$ and the other for the bottom region ${\int <!--\; \int \; -->}_0^m{3}^2-<!--\; -\; -->(3-<!--\; -\; -->x{)}^{2}dy$. The 3 is the outer radius of the washer and the $3-<!--\; -\; -->x$ gives the inner hole of the washer. I can't seem to get the right answer.
$\mathrm{\pi <!--\; \pi \; -->}{\int <!--\; \int \; -->}_m^9(6\sqrt{y}-<!--\; -\; -->y)dy=\mathrm{\pi <!--\; \pi \; -->}(4y^{\frac{3}{2}}-<!--\; -\; -->\frac{1}{2}{y}^2)$ evaluated from 9 to m $=\mathrm{\pi <!--\; \pi \; -->}(\frac{135}{2}-<!--\; -\; -->(4m^{\frac{3}{2}}-<!--\; -\; -->\frac{1}{2}{m}^2)$. Similarly, for the bottom region integral, I get $\mathrm{\pi <!--\; \pi \; -->}(4m^{\frac{3}{2}}-<!--\; -\; -->\frac{1}{2}{m}^2)$. I then try to set the volumes equal to each other and solve for m.
I believe the answer I should get is $\frac{9}{\sqrt[3]{2}}$ but I don't get that value for m.
If I continue, I get $\frac{135}{2}$ =$8m^{\frac{3}{2}}$ - $m^2$ but I don't see any easy way to solve without using a calculator.

How many sides does a regular polygon have if one exterior angle is 72?

Hypotenuse and angle ratio relationship
In triangle $\mathrm{\angle <!--\; \angle \; -->}BAC=90$, $\mathrm{\angle <!--\; \angle \; -->}ABC$: $\sqrt{ACB}=1:2$ and $AC=4cm$. Calculate the length of BC. I am interested in the solution using Pythagoras theorem.

Total area for a natural nested set of convex polygons.
Suppose we have a convex polygon $P_0$ with n given vertices, and we want to "nest" polygons $P_j$ for $j>0$ by taking the midpoints between edges of $P_{j-<!--\; -\; -->1}$ as the vertices. For a regular polygon the total area of all the polygons will form a geometric series that is pretty easy to solve for (at least in terms of trig functions) in terms of the area A of the original polygon $P_0$. However, what if $P_0$ is not regular? If we know the area of $P_0$, is there a formula for the sum of areas of all the nested polygons, in terms of n and the original area? Or do we need more information, like the vertices of $P_0$? If we need to know the vertices, are there formulas in terms of the vertices at least for some small values of n, like $n=3,4$?

Surface integral of angle between fixed vector and the unit normal
S is the smooth solid boundary of an object in R3, and v is its outward unit normal. l is fixed vector in R3. The angle between l and v is $\mathrm{\theta <!--\; \theta \; -->}$. Prove: ${\iint <!--\; \iint \; -->}_S\cos <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})\mathrm{d}\mathrm{S}=0$.

Distance between (5,-6,4) and (4,-1,2)?

A polygon has 6 sides. How would I find the measure of each exterior and interior angle?

Find the perimeter of the triangle with the vertices at (2, 2), (-2, 2), and (-2, -3). I keep getting 18 but my online homework website says it is wrong. Find the point (0,b) on the y-axis that is equidistant from the points (5,5) and (4,−3)?

Find volume under given contraints on the Cartesian plane.
The constrains are given as
$x^2+y^2+z^2⩽<!--\; ⩽\; -->64,x^2+y^2⩽<!--\; ⩽\; -->16,x^2+y^2⩽<!--\; ⩽\; -->z^2,z⩾<!--\; ⩾\; -->0.$
With the goal of finding the Volume. Personally, I have trouble interpreting the constrains in terms of integrals to find the Volume. However, logically z can be maximum of 8 and x or y no larger than 4.

Use an indirect proof to show that if $x^3+x-<!--\; -\; -->1><!--\; >\; -->10$ then $x>1$.

A polygon has n sides. Three of its exterior angels are 70 degrees, 80 degrees and 90 degrees. The remaining ${\displaystyle (n-3)}$ exterior angels are each 15 degrees. How many sides does this polygon have?

Probability circle to be in a circle
We have circle with radius R and inside it we choose at random 2 points A and B.Find the probability the circle with center A and radius AB to be inside the circle(It is possible both circles to be tangent)
What I've concluded : the first point A has range from [0;R] and the second point B must have range from $[0;R-<!--\; -\; -->OA]$ .But how should I find the probability?

What is the distance between (23,-3) and (24,-7)?

What is the distance between (9,-7,1) and (3,-5,-2)?

Conic sections - Hyperbola
If we are given a 2 degree curve equation of a hyperbola, is there a way to find the centre, foci, eccentricity and directrices of a hyperbola , just as we can obtain the equation using foci, eccentricity, and directrix? I searched for it, but only found an answer for ellipse.

Distance between (1,-3,-5) and (-2,-3,4)?

A bicycle wheel with a 5 inch ray rotates ${60}^{\circ <!--\; \circ \; -->}$. What distance has the bicycle traveled?

Probability of finding $n_1$ particles in volume $v_1$ and $n_2$ particles in volume $v_2$.
The formula is:
$P[n\text{ particles in }V]=(\genfrac{}{}{0ex}{}{N}{n})p^n(1-<!--\; -\; -->p{)}^{N-<!--\; -\; -->n}$ where p is $p=\frac{v}{V}$.
I would like to know how this formula generalizes if one wants to compute "the probability of finding $n_1$ particles in volume $v_1$ and $n_2$ particles in volume $v_2$, given that there are N indistinguishable particles in total, that the total volume is V and that $v_1$ and $v_2$ are disjoint."

What is the measure of an interior angle of a regular polygon with 20 sides?