Let the radious of the circle is r. Now if i want to find out the area of the crosssection ( mentioned by red color) then how to proceed?
119819
Suppose you have 2 circles that intersect each other in such a way that each circle passes through the other's center. What is the area between the circle(or common area) i.e. area between the centres of the circles?
57830
Given three vectors $u,v,w \in S^2$ and the triangle $[u,v,w]$ I want to find its circumscribed circle. However, I don't know how to approach this problem. Would some one please explain?  In my understanding, one needs to somehow find the altitudes of the spherical triangle and then the point of its intersection (circumcentre)? Then what about the radius?
104153
In the series of concentric shapes, this is another drawing that I'm unable to figure out. The top is not quite a complete semi-circle. Note that some of my labels got left out in the scan.
50835
Question:  Consider the following square sheet of metal (grey) with the side s = 10 cm. From it, we want to cut the parts of a cylindrical can with a lid and a bottom (bblack). Determine the radius and height of the can when the volume is maximal.   The rectangle and circles have to be cut in entire, singular pieces, so no trickery to minimize waste and no diagonal placement of the rectangle.  Solution has to use optimization / differentiation.  Attempted solution:  In order for this to work out the circumference of the circles and one side of the rectangle has to be identical. Otherwise they will not fit together in a cylindrical can.  This leads us to two cases that needs to be treated separately.  Case 1: The rectangle is standing upright.  Case 2: The rectangle is lying on its longest side.  There are two cases that needs to be considered separately.  Case 1:  The right-hand side of the above image must be the circumference of a circle, so  $$s_1 = 2r\pi$$  The bottom-side must be the height of the cylinder plus two radii of the circles:  $$s_1 = h + 2r$$  Putting these together becomes:  $$2r\pi = h + 2r = 10$$  Solving for h:  $$h = 10 - 2r$$  The volume of the can:  $$V = r^2\pi h$$  Adding in the earlier formula to get rid of h:  $$V = r^2\pi h = r^2\pi (10 - 2r) = 10r^2\pi - 2r^3\pi$$  Taking the derivative:  $$V' = 20\pi r - 6\pi r^2$$  Setting derivative equal to zero:  $$V' = 0 \Rightarrow r = \frac{10}{3}$$  (neglecting trivial r = 0 solution)  Height is then:  $$h = 10 - 2r = 10 - \frac{20}{3} = \frac{10}{3}$$  But this is not really possible given the side of the metal sheet cannot be more than 10 cm and $$2r\pi &gt; 10$$.  Case 2:  The bottom becomes:  $$2r + 2r\pi = 10 \Rightarrow r= \frac{5}{1+\pi}$$  The right side becomes:  $$h = 10$$  However, none of these answers match the solution:  $$r = \frac{5}{\pi}$$  $$h = 10(1-\frac{1}{\pi})$$  Did I get the starting point for the possible cases right? Where did I go wrong? How come it appears that no calculus is needed in the second step?
48214
What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane?   One approach is to find the length of each side from the coordinates given and then apply . Is this the best way possible?   Is it possible to compare the area of triangles with their coordinates provided without actually calculating side lengths?
102178
I have coordinates of 3d triangle and I need to calculate its area. I know how to do it in 2D, but don't know how to calculate area in 3d. I have developed data as follows.  (119.91227722167969, 122.7717056274414, 39.3568115234375),  (119.8951187133789, 122.7717056274414, 39.38057327270508),  (121.11941528320312, 123.2818832397461, 38.41301345825195)
97061
I'm not sure how to solve this problem. Can you guys provide some input/hints?      Let $A=(x_1,y_1)$, $B=(x_2,y_2)$ and $C=(x_3,y_3)$ be three points in $\mathbb{R}^{2}$.      Show that the area of $\Delta ABC$ is given by   $\frac{1}{2}\left| \det(M) \right|$, where   $$M = \begin{pmatrix} 1 &amp; 1  &amp; 1\\  x_1 &amp; x_2 &amp; x_3\\  y_1 &amp; y_2 &amp; y_3 \end{pmatrix}$$
51856
Find the radius of the inscribed circle of a right triangle. The triangle's height is $\sqrt{6} + \sqrt{2}$ while the bisector of the right angle is 4.  This seemed like a generic similar triangles problem but it isn't that simple when I tried to solve it. Any help?  EDIT: Here is a picture of the problem
31323
So, my dilemma here is... I have an axis. This axis is given to me in the format of the slope of the axis in the x,y and z axes.  I need to come up with a parametric equation of a circle. This circle needs to have an axis of rotation at the given axis with a variable radius.  I've worked on this problem for days, and still haven't come up with a solution.  I'm using this circle to map the path of a satellite, programmed in C.  And help would be greatly appreciated.  Thanks!
87755
Given the length ot the sides $a , b$ and $c$  of $ \triangle ABC$.  What is the length of the radius of the circumcribed circle?  After some formula substitution I came to the monster formula:  $$ \frac {a b c}{\sqrt{2a^2b^2+2a^2c^2+2b^2c^2-a^4-b^4-c^4}} $$  Can this formula be simplified?
111068
Among all triangles inscribed in the unit circle, how can the one with the largest area be found?
87504
How to find the expected area of the intersection of a set of triangles made up $N$ random points that are picked uniformly inside a circle? The triangles must contain the origin of the circle. If none of the trinagles contain the rigin we pick again $N$ random points until at least one triangle contain the origin and calculate the expected intersection area of all the triangles that contain the origin.
101483
I have a psd that has rounded rectangle. I need to create a new rounded rectangle with the same radius. How do I determine the radius used on that said rounded rectangle?   I feel like this is such an awful question but thanks in advance for answering.
81971
I have to find $r'(t)$ and $||r'(t)||$ for $r(t)=&lt;t,t,t^2&gt;$, which I know how to do.  $r'(t)=&lt;1,1,2t&gt;$ $||r'(t)||=\sqrt{2+4t^2}$  The problem is that my professor didn't explain how to find where each curve hits the $x-y$ plane (if it hits).  I'm guessing that I have to make $z=0$, but where?
57468
So last night, my D&amp;D group ran into an interesting question. The spell Entangle says:     Area: plants in a 40-ft radius spread.    We (like a lot of people, I think) play D&amp;D on a square grid, where each square represents a 5'x5' area. In that context, how do you find the area when you're given a radius?   The books seem unclear, probably since the definition of a radius is fairly simple. My suggestion was to treat it the way you treat range: start at the center, and any square you can get to with 8 five-foot 'steps' is affected. Someone else thought it should be a square that was 80'x80'; that is, a 'circle' that's 80 feet across, since a lot of other 'circles' in D&amp;D are actually treated as squares. That's a simply huge area, though -- bigger than the board we use. We wound up using a 40x40 square, since my counting would take more time and the 80x80 square was huge. 40x40 was still big enough to encompass the entire fight.   My question: how does a radius measurement (and by extension, circles in general) work on a square grid?
110600
I am trying to solve this question but i cannot seem to find the center (h,k). I know i have to use the equation of circle form '(x-h)^2+(x-k)^2=r^2' and after that form to solution of the 2 point given and simultaneous equations them. I got the answer for that which is h+k=2 and that is where i got stuck. Please anyone help me with this. i also post my working as well.
5529
Given the radius and $x,y$ coordinates of the center point of two circles how can I calculate their points of intersection if they have any?