Area is a quantity that expresses the extent of a two-dimensional or three-dimensional surface or shape.

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A scalene triangle with no right nor obtuse angle

I want to find the area of the perfect triangle, i.e. a triangle with no particularity whatsoever : no side shall be equal to another, no right angle, no obtuse angle. So I gave myself a segment $[...
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30 views

What will be area of an equilateral triangle? [on hold]

my question is A side of an equilateral triangle is 24 root 3. Inside this triangle two other equilateral triangles is made such that there inner areas becomes same. find out side of smallest ...
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2answers
38 views

Area calculate, ln [on hold]

Hey can someone help me with this area calculation? I need to show that the area between the lines $y= 1/x$ and $y= x/2$ and $y= 2x$ equals to $ln 2$. Thanks!
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2answers
40 views

Area of the polar figure enclosed by the circle $r=2$ and the cardioid $r=2(1+cos θ)$

This is exercise 7, of the book Engineering Mathematics by Stroud, Chapter 24, Further Problems section. Here's a graph i made of the figure as i see it: It gives the answer as $π+8$. The integral ...
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37 views

Find the area of $4x^2-2xy+y^2=1$ [closed]

Any help? Ive tried everything I can think of.
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1answer
29 views

Why is inradius $\times$ surface area equal to thrice the volume?

"Inradius" means radius of largest sphere that is tangent to all faces. For example: Cube - Surface area $= 6a^2$, Inradius $= a/2$, Volume $= a^3$. Sphere - Surface area $= 4\pi r^2$, Inradius $= ...
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2answers
26 views

Area between two polar curves method

The question is not too hard. I sketched them and they were correct which was not too bad. I then did the second part by finding the intersection points between the two curves which are $\frac{\pi}{...
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2answers
34 views

Deriving surface area of a sphere from the circumference

given the circumference of a circle, which is 2πr, how many times do I have to add it to itself to cover a whole surface of a sphere and deriving 4πr^2?
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2answers
37 views

What would be the area of this Red Marked points? And how to calculate this?

I have been given the length $L$ and the width $W$ of a rectangle and the radius $R$ of circle which is situated in the center of the rectangle . I need to find the area of the red marked portion. ...
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1answer
43 views

What is the area of triangle ABC?

Verbatim my Math test- Consider a polynomial $y=P(x)$ of the least degree passing through $A(-1,1)$ and whose graph has two points of inflexion $B(1,2)$, and $C$ with abscissa 0, at which, the curve ...
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3answers
60 views

How to find the area of the following isosceles triangle

I am stuck with the following problem : What is the area of an isosceles triangle whose equal sides are $20$ cm and the angle between them is $30^{\circ}$ ? It is a nineth standard problem and ...
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2answers
129 views

how is it that $\int_0^x 2\pi r\ dr$ is equal to the area of a circle [closed]

I'm studying calculus and I'm having some basic questions, this one is regarding the area of a circle. we know, from some guy, that the circumference of a circle is $2 \pi r$ and the area can be seen ...
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1answer
30 views

Area in terms of $x$.

A wire of $80 \, \mathrm{cm}$ is arranged to form $3$ sides put against a wall forming a rectangle. The longest sides of the rectangle is the wall and a piece of wire with length $x \, \mathrm{cm}$. ...
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1answer
23 views

Area included in the graph of various functions

I've some problems recognizing which one is the area between various function. In this case i need to calculate the area between 3 lines and a curve, exactly between: $x+3,x^2-9,x=5 ,x=0$ I can't ...
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0answers
26 views

Finding area of hypocycloids (without integration)

I have been trying to find the area of hypocycloids, I understand how to do it with integration. But the thing is I wanna find some other method for finding its area. In one of the sites online, I ...
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1answer
29 views

Factoring Polynomials: How do I express the area and perimeter in factored form?

Our topic is factoring polynomials, and I can't seem to solve this question: Express the area and perimeter of the shaded region in factored form. We've discussed how to solve for the ...
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1answer
43 views

How do I find the partial derivatives of heron's formula?

Heron's formula finds the area $A$ of a triangle with sides of length $a$, $b$, and $c$: $$A=\sqrt{s(s-a)(s-b)(s-c)}$$ where $s$ is the semiperimeter of the triangle: $$s=\frac{a+b+c}{2}$$ How do ...
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1answer
55 views

Determine the varying area of the shaded region

How to estimate the area of the shaded region shown in the attached Figure? Note that in the figure, $p$ has a maximum and minimum values of $p_a$ and $p_b$ respectively. Moreover, $p$ follows a ...
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1answer
25 views

Why can the determinant of a transformation matrix, and the original are be used to find out the new area?

Im studying Year 11 Mathematical Methods. Within the book there are several questions which give vertices (or sometimes just the area) of a particular shap, plus a transformation matrix to be applied ...
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2answers
110 views

Dividing a circle into $3$ equal pieces using $2$ parallel lines

I originally found this question in James Stewart's Calculus, specifically in one of the Problems Plus sections. The question asks how $3$ people can share a pizza while making just $2$ cuts, ...
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1answer
55 views

Determine the largest area of an ellipse enclosed by the hyperbolas ($xy=1$ and $xy=-1$)

Question: An elipse with equation $$ {x^2\over a^2} + {y^2\over b^2} = 1 $$ is enclosed by the hyperbolas given by $xy=1$ and $xy=-1$. , Determine the largest area of an ellipse enclosed by the ...
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3answers
45 views

Area of subset of $\mathbb{R}^2$

Determine the area of $$ \{(x,y) \in \mathbb{R}^2: x^2 + y^2 \le r^2\}, (r \in (0, +\infty))$$ My problem is to get the correct definite integral. I think the limits are $0$ and $r$ because $r \in (...
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2answers
23 views

Calculating Open ended cone height

If I have the diameter for both bases of the open-ended cone, and they are 20, and 16. Is there any way I could get the height of that open-ended cone?
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3answers
88 views

Find the area between $y = -x^2+4x$ and $y = -x +4$.

I'm a high schooler. I'm studying for an exam and got stuck with calculating the area between two functions. Picture of the question : (1) $y = -x^2+4x$ (2) $y = -x +4$ $A(1,5), B(4,0)$ ...
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62 views

The fraction of the larger hexagon that is shaded?

This is from Australian Maths 2013. In a regular hexagon,the midpoints of the sides are joined to form he shaded regular hexagon.What fraction of the larger hexagon is shaded? Since the larger ...
2
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1answer
35 views

How to find this area of intersecting curves using integration?

I have to find the area of the yellow part given $ABCD$ is a square of $10cm$ side and the curved lines that run diagonally are the circumference portions of the quadrants of circles with each side of ...
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1answer
36 views

Squares and areas

In the dotted sheet below, the distances, both horizontally and vertically, between every two neighbouring points are equal. Ann drew all possible squares by connecting four of the points. How many ...
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20 views

Average distance for 2 random points

An n-dimensional region $\Omega$ is given. Randomly two points on the region are chosen. These points have an uniform distribution. Is it possible to calculate the average (=expected) distance with ...
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1answer
22 views

How do I find the partial curved surface area on a hemisphere?

Want to develop a general formula to find the surface area of the highlighted area for any radius (say) r for the hemishphere On spreading out this surface on a plane it can be measured with help of ...
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2answers
73 views

Evaluate $\iint { \sqrt{\left| y-{ x }^{ 2 } \right|}\, dx\,dy } $ over a rectangle

Question: I want to evaluate $\iint_R {\sqrt{ \left| y-{ x }^{ 2 } \right|}\, dx\,dy }, $ where $R=[-1,1]\times[0,2]$. Indeed $x\in[-1,1]$ and $y\in[0,2]$. My approach: Since, $|y-x^2|$ is positive ...
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area of a rectangle

I know that to use polar coordinates instead of $dxdy$ we have $dA=rdrd\theta$. As such, we can have a double integral like $$ \int_{\theta=a}^{\theta=b}\int_{r=c(\theta)}^{r=d(\theta)}f(r\cos\theta,r\...
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2answers
100 views

Does this alternating sum of roots converge to $\sqrt2$?

This problem arose from what I'm hesitant to call an investigation into a certain type of "quadrature". Starting with the unit disk, I partition it into $p$ pieces by cutting the disk with vertical ...
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3answers
26 views

Find the area of the region $y=2x$, the $x$-axis, lines $x=1$ and $x=4$

Find the area of the region $y=2x$, the $x$-axis, lines $x=1$ and $x=4$ Here's what I did: $$A = \lim_{n \to +\infty} \sum^{n}_{i=1}f(x_{i-1})\Delta x \\ = \lim_{n \to +\infty} \sum^{n}_{i=1} 2(i-1)...
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28 views

Filling a pool.

i have a pool of 2 meter radius, and 1.5 height. The pool is a perfect Cylinder. How many bottles of 1 liter i have to spill into the pool, so it will be completely full. The Volume of the Cylinder is ...
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1answer
32 views

Area of all triangles involved in a big triangle.

I have a big triangle made up of several small triangle as depicted in picture given below. Suppose, there is one generic triangle of this shape which is formed by joining points arranged in n rows....
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24 views

Sum of Area of Circles. [duplicate]

A circle of radius x cm is inscribed in an equilateral triangle and is tangent at three points. Three smaller circles are inscribed so that they are each tangent to two sides of the triangle and to ...
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2answers
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In the figure below,three congurent semicircles with centres P,RQ,R are drawn on each side of three equilateral triangle.Find shaded part's area?

In the figure below,three congurent semicircles with centres P,RQ,R are drawn on each side of three equilateral triangle.Find shaded part's area?
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1answer
16 views

Find equal side lengths for isosceles triangle from middle angle and area?

I know that this is a really easy question, but I am looking for the answer to this question: The area of this isosceles triangle is 5cm squared. The angle ABC is 22 degrees. Work out ...
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2answers
48 views

Point A is picked randomly in a circle with a radius of 1, and center O. What is the variance of length OA?

Point A is picked randomly in a circle with a radius of 1, and center O. What is the variance of length OA? I believe the CDF has to found first, then we need differentiate it, find the expected ...
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27 views

$N$-dimensional volume (of revolution)

Consider the system of coordinates $\{x_{1},x_{2},...,x_{n}\}$ and an n-dimensional shape such that, in $\{x_{1},x_{n}\}$ (and $x_{2}=x_{3}=...=x_{n-1}=0$) it is inside the lines $x_{n}=ax_{1}+b$ and $...
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1answer
44 views

Find radius of circle (or sphere) given segment area (or cap volume) and chord length

The goal is to design a container (partial sphere) of given volume which attached to a plane via a port of a given radius. I believe this can be done as follows but the calculation is causing me ...
2
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1answer
36 views

Calculate Section (area) of N-Dimensional Tube

I have the following n-dimensional shape $1=\sum_{i=1}^{n}a_{i}x_{i}^{2}$ where $a_{i}>0$ and I'd like to calculate the cross-section area inside. Any suggestion? Note: I call it an $n$-...
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Find the average of points/areas in a chart

I've calculate areas at certain points along X. The X axis is along a beam and these areas are the required area of steel at ...
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2answers
39 views

Find the area of $S=\{(x,y)|\rm{\exists ~}\theta,\beta,x=\sin^2{\theta}+\sin{\beta},y=\cos^2{\theta}+\cos{\beta}\}$

Let $S$ be the domain defined by $$S=\{(x,y)|\rm{\exists ~}\theta,\beta,x=\sin^2{\theta}+\sin{\beta},y=\cos^2{\theta}+\cos{\beta}\}$$ find the area of $S$ This is middle school problem,so I think it ...
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1answer
21 views

Spaces relations

I have a physics question for which I need to determine the radius of a circle. Given are two Ellipse shapes with the same center (0,0 in a Cartesian space). the Height and Width of the smaller is 1[...
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1answer
44 views

Calculate the area of a sphere drilled by two cylinders.

Let $S$ be the sphere given by the equation $x^2+y^2 +z^2 =4$ cut with $z \geq 0$. Now, we drill the semisphere that is left with two vertical cylinders of radius $1$, whose axes are respectively ...
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1answer
95 views

Find Minimum area of given hexagon. Geometry Question.

It has been a year Since I am searching for an answer to this question. This question was probably asked in International Mathematics Olympiad but I am not a 100% sure. Q : ABCDEF Is a concave ...
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1answer
20 views

The Area of Card Required to Make a Cone

The solution of the question I'm trying to understand is this: The part of the solution I don't understand is "the area of card needed for each hat (cone) is $\frac{1}{2}r^2$", because I would ...
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2answers
74 views

Compute the area of a oval based 2d geometry

I know that the area of a shape generated as below $R=r_0+a_1\cos(\theta)+a_2\cos(2\theta)+a_3\cos(3\theta)+...$ Where you can plot it and see the area value in matlab by: ...
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1answer
52 views

Area bounded by the x-axis and the curve..

Find the area bounded by the $x$ axis and the curve $$y = \frac{(x^2−x−2)}{(x^3+8)}$$ between its points of intersection with the $x$ axis. So point of intersection are $(-1,2)$ since $$f(y)=0$$ when ...