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Questions tagged [area]

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

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optimization exercise to find the maximum area of a rectangle formed by two rectangles with a line. Literal exercise.

A rectangle R in the plane has corners at (+-8, +-12), and a 100 by 100 square S is positioned in the plane so that its sides are paralleul to the coordinate axes and the lower left corner of S is on ...
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What is the area of this figure?

I want to know how to find the area of this shape: Yellow, white and blue shapes are ellipses. Red is a square. The blue ellipse is not cut in half by the square. I know that I have to add up all ...
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Area of a circle segment on sphere, given radius (meters) and central angle (degrees)

Situation I have a circle segment and some information about the circle it belongs to. Given Information: radius of the circle in meters central angle in degrees lat/long of all three points on the ...
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How to double the circle?

I'm looking for a compass-and-straightedge method to construct a circle that has area twice of the area of another circle, with no prior knowledge of π, without knowledge of the formula for the area ...
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How do I find the area bounded by a $y=\frac{4}{3}x^2+\frac{12}{3}x-3$ and $y=\sqrt{x}$

I am having difficulties with this problem:$y=\frac{4}{3}x^2+\frac{12}{3}x-3$ and $y=\sqrt{x}$. Graphng two of the functions I get the following: Graphing both functions shows that they intersect at $...
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how to calculate wall thickness of a mesh?

based on similar questions on mesh volume, volume of a mesh can be calculated by following equation: volume = ((vec1 x vec2) . vec3) /6 where vec1, vec2, and vec3 are the vectors from origin to a ...
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Determining the area of a right triangle, perimeter given, hypotenuse value given in terms of one of the legs.

The problem states: Right Triangle- perimeter of $84$, and the hypotenuse is $2$ greater than the other leg. Find the area of this triangle. I have tried different methods of solving this ...
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Find region of $0\le y\le1$ and $y\le x\le1$ [on hold]

How i can find the region that is bouned from these inequalities? Any general rule when we have to deal with these inequalities? Ploting the graphs?
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Function whose graph is dense in the plane [duplicate]

Is there a function $f:\mathbb R\to\mathbb R$ such that for every disc in $\mathbb R^2$ the graph of that function has at least one point that lies inside that disc? I searched for something similar ...
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Help find work done to pump fluid

A tank is full of oil weighing 20 lb/ft${^3}$. The tank is an inverted right rectangular pyramid (with the base at the top) with a width of 2 feet, a depth of 2 feet, and a height of 5 feet. Find the ...
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What is the area of the triangle given its sides? [on hold]

What is the area of the traingle with sides 2, 3, and 5?
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Orthogonal projection of an ellipsoid

Suppose an ellipsoid given by $\{{(x,y,z)| {x^{2}+\frac{y^{2}}{4}+\frac{z^{2}}{9}}}=1\}$, find the area of the orthogonal projection of the ellipsoid on the plane ${2x+4y-5z=10}$. What is the right ...
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How to determine the area of a rotated ellipse?

The ellipse $6x^2+4xy+5y^2+8x+8y+1=0$ is neither expressed in terms of $x$; like $y=\pm\sqrt{a^2-x^2}$, nor in terms of $y$; like $x=\pm\sqrt{a^2-y^2}$. Separation of $x$ (or $y%$) may be impossible. ...
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How to calculate the area and volume of a random 3d shape knowing only its coordinates?

Given a set of 3d points which make up an vector object of any shape (with any number of points), without the edges being known, how can the object's edges be found/detected so that the object's ...
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Area that is bounded by functions

The functions $$f_k(x)=\frac{x+k}{e^x}$$ are given. Let $A(u)$ be the area that is bounded by $f_1, f_3$, the $x$-axis und the line $x=u$. I want to check the area if $u\rightarrow \infty$. $$$$ ...
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The surface area of a 3d rectangle without a top. The length is 1 the width is 4 and the height is 6 [closed]

The length is 1 the width is 4 and the height is 6. I have tried to find the answer but i can't.
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Calculate Overlapping Area of $2$-Dimensional Shapes

I am running a Computer Simulation where 2 Shapes are moving towards each other and will eventually overlap. I want to calculate the overlapping Area of the shapes - in this example a Circle and a ...
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Find area of quadrilateral in triangle. [closed]

What is the area of $HIJK$ quadrilateral, if the area of $ABC$ triangle is $70$, $BE=ED=DA$, and $BF=FG= GC$?
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Finding percentage increase of area [closed]

Find the percentage increase in the area of a triangle if its each side is doubled?since no information is given about the type of triangle in question, so should i take a equilateral or isosceles or ...
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Do the given perimeter and area corresponds to many shapes? [closed]

I have a perimeter P and area A of a planar shape. How to prove that there are many shapes that corresponds to those perimeter and area values?
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How do you calculate area of two circles joined by tangent lines

please can you help to provide the mathematical steps required to calculate the area of a shape formed by two circles of different diameter joined together by two tangential lines.
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Area of the region

Let $O(0,0), A(3,0), B(3,2)$ and $C(0,2)$ be four vertices of a rectangle. Let $$d(P,OA)≤\min {\Bigl(d(P,AB),d(P,BC),d(P,OC)\Bigl)}$$ where $d$ denotes the distance of the point $P$ with the line ...
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Area in d dimensions of a spherical part [duplicate]

Let $S^{d-1}$ be the unit sphere (centered at $O$) in $d$ dimensions. One can show that when $d=3$, for fixed $x\in S^{2}$ the area of $P(x) = \{y\in S^{2}, \angle xOy < \alpha \}$ is $2\pi(1-\cos\...
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Regarding Areas of volumes

I have a question: In general, area of square is Length times Length. If the length is less than $1$ then do we end up with less area? Like if we have a side whose length is $0.5\mathrm{m}$. Then ...
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BdMO - 2018 Regional - Geometry 9 [on hold]

In $ABCD$ tetragon $E $ and $F $ are mid points of $AB$ and $AD$ respectively. $CF$ intersects $BD$ at point $G$. If $\angle FGD = \angle AEF$ and the area of $ABCD$ is $24$ , what is the area of $...
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1000cm^3 of metal is to be cast as a rectangular block with square ends (Calculus Optimization question)

1000cm^3 of metal is to be cast as a rectangular block with square ends. Use calculus to show that for the least surface area a rectangular the rectangular block needs to be a cube. Having some ...
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Area of a triangle in space using determinants from 3 points

I know there is a way to find the area in plane using 3 points but when it comes to space(3D) does it work too? if so how should it look? Thanks
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1answer
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Calculate surface normal and area for a non-planar quadrilateral

Given the four coordinates of the vertices, what is the best possible approximation to calculate surface area and outward normal for a quad? I currently join the midpoints of the sides, thus ...
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1answer
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Calculating integral over unit ball using co area formula

Calculate $\int_V\frac{1}{2+(x^2+y^2+z^2)^{\frac{3}{2}}}dxdydz$, where $V=\{(x,y,z)|x^2+y^2+z^2\lt 1\}$ using the co area formula. So I know that the formula is: $\int_V f dx$ = $\int_a^b\int_{M_c}\...
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find the area of the octagon [closed]

Find the area of the colored octagon (tell the ratio between square $ABCD$ and the colored polygon). Square $ABCD$ is a perfect square, and $E,F,G,H$ are the midpoints of the line they are ...
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Overlapping inscribed triangles.

Question: Let $T_1$ be the equilateral triangle inscribed by the unit circle centered at the origin of $\mathbb{R^2}.$ Now let $T_2$ be the triangle induced by clockwise rotating each vertex of $T_1$ ...
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What to do further in Find the maximum area of ellipse…

$$Question$$ Find the maximum area of ellipse that can be inscribed in an isosceles triangel of area $A$ and having one axis along the perpendiculur from the vertex of the triangle to the base. $$...
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Antiderivative: does it represent any area?

I was told that an antiderivative doesn't represent any area, it is just a family of functions, but today I noticed something quite interesting thing: If we have $f(x)=2x$, then $F(x)=x^2+C$. The ...
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Area of Semi Circle With an Inscribed Triangle

Suppose we have a triangle with a right angle at its height, with side a, 10 inches, side b, unknown, and side c, 24 inches; inscribed in a semi-circle. Now, I'm asked to find the area of the semi ...
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Calculate the ratio of the sides of a given triangle given the ratio of areas.

Given a triangle $\triangle ABC$, points $M$, $N$, $P$ are drawn on the sides of the triangle in a way that $\frac{|AM|}{|MB|} = \frac{|BN|}{|NC|}= \frac{|PC|}{|PA|}=k$, where $k>0$. Calculate $...
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Surface Area of a revolution

So I had a question involving Surface Area. Find the surface area generated by revolving the curve $x = y^3$ about the $y$-axis for $0 \leq y \leq \sqrt[4]{11}$. (a) $23\pi\quad$ (b) $37\pi\...
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Area of Overlapping Circles a Convex Function?

All, I'm not sure how to phrase this. I've calculated the area of overlap of two circles of radius 1, with the first circle centered at $x=0$ and the second at $x=2d$ as $$A = 2\arccos(d) - 2d\sqrt{1-...
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find minimum value of required [duplicate]

let $ f$ be a continuously differentiable real valued function on $[0,1]$ such that $ \int_{1/3}^{2/3} f(x) =0 $. find minimum value of $\frac { \int_{0}^{1} f'(x)^2 dx }{ (\int_{0}^{1} f(x)dx)^{2} }...
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Closed-form representation of area below a function

Let $f:[0,1]\rightarrow[0,1]$. $f$ is strictly increasing in $x$ if $x\in[0,x^*]$ and strictly decreases otherwise. Suppose that I'm interested in finding the area of the domain where $f(x)\leq t$. ...
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How to calculate area of an ellipse based on its formula?

How can I determine the area of a half-ellipse if all that is given is $y = \sqrt{1-n^2x^2}$? I have tried both geometry and calculus, but without convincing results… Thank you
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1answer
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Compute area of triangle

Problem Triangle can be formed in between three points. These three points are in $\mathbb{R}^3$ and in my case these points are: $p_1=(0,4,6),p_2=(-5,3,1),p_3=(2,1,2)$. Compute area of this ...
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How to find volume and surface area of a spindle torus?

I know that you can use the formulas described in Pappus' centroid theorem, detailed here. But does Pappus' centroid theorem hold true for all forms of a torus: ring, horn, and spindle? I found ...
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The Jacobian of $(x,y)\mapsto (x+y^2,y+x^2)$ under the substitution $u=x+y^2$ and $v=y+x^2$.

I am given the map $(x,y)\mapsto (x+y^2,y+x^2)$. I am unable to find the Jacobian by making the substitution $u=x+y^2$ and $v=y+x^2$. Any hints would be appreciated. (I am trying to find whether the ...
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1answer
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Deriving the formula of the surface of a sphere using triangles.

My friend tried to find the formula of the surface of a sphere using the following reasoning, but we can't see the mistake: Let's first take half a sphere and divide the sphere into infinitely small ...
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1answer
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Area under the graph of $r\mapsto\binom nr$

The question: Given $n$ is a natural number and $r$ is varying from $0$ to $n$, find the area under the graph of $r\mapsto\binom nr$, taking the $\Gamma$-function definition of factorial. ...
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1answer
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Why is the surface area of a sphere equal to $4\pi r^2$ [duplicate]

I have absolutely no idea where that formula comes from, considering the fact that I am a fifteen year old. According to me, one way to think of it is to arrange $4$ circles having radius equal to ...
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2answers
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Area of triangle using double integrals

I have one (rather simple) problem, but I'm stuck and can't figure out what I'm constantly doing wrong. I need to calculate area of triangle with points at $(0,0)$, $(t,0)$, $(t,\frac{t}{2})$. In ...
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Area of a triangle inside an ellipse

$F_1$, $F_2$ are are foci of the ellipse $\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$. $P$ is a point on the ellipse such that $|PF_1|:|PF_2|=2:1\;$, then how could I figure out of the area of $∆PF_1F_2$? As ...
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Green’s theorem double into line integral

How I can convert this double integral into line $\int \int \frac{dx dy}{y^2} $ we have $\frac{dQ}{dx}-\frac{dP}{dy}=\frac{1}{y^2}$ how to find $Q, P$ functions? Any two functions works? $P=1, Q=\frac{...
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Is it possible to integrate something that isn't a function?

How would you "find the area under the curve" of something that isn't quite a curve? The graph may be curved at places, but if it's not a function (the same x value has more than one corresponding y ...