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Questions tagged [area]

Area is a quantity that expresses the measurement of the extent of a two-dimensional shape.

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Finding the area of triangle DEF in triangle ABC with given side lengths and specific points D and E [closed]

Let ABC be a triangle with AB = 18, BC = 24, and CA = 20. D is placed on AB such that AD = 15. E is placed on BC such that EC = 20. Call the intersection of lines AE and DC point F. Given the ...
Sebas Domenech's user avatar
1 vote
1 answer
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What does $\mathrm{d}(x^2)$ mean in the $\int_a^bf(x)\, \mathrm{d}(x^2)$ or is such a thing not valid

Here is my understanding of the $\int_a^bf(x)\, \mathrm{d}x$: \begin{align} \int_a^bf(x)\, \mathrm{d}x = \lim_{n \to \infty} \frac{b - a}{n} [f(a) + f(a + h) +... + f(a + (n-1)h)] \end{align} where $h ...
ADITYA VIKRAM SINGH's user avatar
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Finding area of triangle in Argand Plane

Given $|z|^2=4$, find the area of the triangle formed by the complex numbers $z$, $\omega z$, $z+\omega z$ (here, $\omega$ is the complex cube root of unity). I understand the solution which claims ...
Satyam sharma's user avatar
1 vote
1 answer
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Find the ratio of the area of ​the triangle $PRQ$ ¸and the area of ​the triangle $ETA$

the problem On the side $BC$ of the triangle $ABC$, the points $D$ and $E$ are considered such that $BD = DE = EC$. Let $M$ the middle of the segment $AD$, $BM ∩ AE = {P}, CM ∩ AE = {Q}$. $RM$ ¸and $...
Pam Munoz Ryan's user avatar
0 votes
1 answer
57 views

Closed form for the area under $ f(x):=\lim_{N \to \infty}\frac{\pi(Nx)}{\pi(N)} $

Define a function $f:\Bbb Q \to \Bbb Q$ by the following $$ f(x):=\lim_{N \to \infty}\frac{\pi(Nx)}{\pi(N)} $$ where $\pi(\cdot)$ is the prime counting function and $N\in \Bbb N.$ I would like to find ...
zeta space's user avatar
0 votes
2 answers
86 views

Area with double integral in polar coordinates

Determine the area interior to $y^2=2ax-x^2$ and exterior to $y^2=ax$. The area in artesian coordinates is $$\int_{0}^{a}\int_{\sqrt{ax}}^{\sqrt{2ax-x^2}} dydx$$. To convert it into polar coordinates ...
a_i_r's user avatar
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1 answer
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What is the fault in this method of finding second moment of area of a circle

I am trying to find the second moment of area of a circle about a diameter using first principles. Place the centre of the circle at the origin of XY-plane. Now consider a tiny circular sector with an ...
Jarvis's user avatar
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3 votes
0 answers
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Determining the significance of a curve's factors

Given the equation $x^2+x+1$ you could easily determine that $x^2$ will have the greatest overall impact on the curve--then $x$ and finally $1$. And this holds true for any coefficients present as the ...
SlavaCat's user avatar
0 votes
1 answer
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Lebesgue outer measure in $\mathbb{R}^2$ in terms of a grid of $h$-squares

For a set $D\subseteq\mathbb{R}^2$, the Lebesgue outer measure of $D$ is defined by $$\lambda^\ast(D)=\inf\bigg\{\sum_i\lambda(I_i)\mid D\subseteq\bigcup_iI_i\bigg\},$$ where $\{I_i\}$ is a sequence ...
ashpool's user avatar
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2 votes
1 answer
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The area of a inscribed polygon tends to the area of the circle

As it is broadly known, given a circle of radius $r$, its area is equal to $\pi \cdot r^2$. My goal is to prove this formula using inscribed polygons. Let´s call $n$ the number of sides of a regular ...
IkerUCM's user avatar
  • 402
2 votes
2 answers
191 views

Area swept by the circumference of an ellipse as it slides such that it is always tangent to the $x$ axis at the origin

You're given the ellipse $\frac{x^2}{a^2} + \frac{(y - b)^2}{b^2} = 1,$ for known $a$ and $b$. Now you slide the ellipse and rotate it such that it remains tangent to the $x$ axis at the origin all ...
that's what it is's user avatar
2 votes
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The measure of what sets is uniquely determined by _finite_ additivity (and translation invariance and normalisation)?

I am very familiar with measure theory but am currently wondering about how far finitary methods can take you. Two aspects have to be differentiated: the unique determination and the calculation of ...
justanotherhumanbeing's user avatar
2 votes
0 answers
33 views

What is the maximum area of n non-overlapping equal area triangles inscribed in a circle of radius

What is the maximum area of n non-overlapping equal area triangles inscribed in a circle of radius 1? For n = 1, the triangle is equilateral. For n = 2, we have 2 isosceles right triangles sharing a ...
Ultima Gaina's user avatar
8 votes
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probability of two confined randomly walking bodies overlapping

EDIT: I have tried to rephrase the problem, title, and context to my solution I am wondering about expanding a problem I have to the continuous domain. The problem is defined as such: Problem Given $N$...
gokudegrees's user avatar
-3 votes
0 answers
47 views

Area of a quadrilateral with only sides [closed]

Well, this is very simple Question but I am not good with Maths. Can someone suggest me what is the area of an irregular quadrilateral with sides measurement as A, B, C and D. Here A < B < C <...
user avatar
0 votes
2 answers
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Area of the parallelogram formed by joining the midpoints of the sides of a quadrilateral

$E, G, F$ and $H$ are the mid-points of the sides of the quadrilateral $ABCD$. Prove that the area of $EGFH$ is half of the area of $ABCD$. Since the sides of $EGFH$ are parallel to the diagonal of ...
Soheil's user avatar
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4 votes
4 answers
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I need help in understanding the alternative solution provided to solve this geometry question of calculating area of quadrilateral

Question: Solution provided: I understand this part that equal chords of a circle subtend equal angles at the center, but after this the faculty transformed this whole diagram to one shown below in ...
Vasu Gupta's user avatar
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Prove that the Hausdorff distance and Area metric are not equivalent on the set of all bounded plane polygons.

Prove that the area metric, $d_{\Delta}$, is not equivalent to the Hausdorff distance between two sets. The book and definitions are here [1] (4.Dx & 4.Ex). The approaches I’ve tried are here: Let ...
Rutvaj Nehete's user avatar
4 votes
3 answers
134 views

Find area of figure and volume of body

Problem Find area: $$\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)^2=\frac{xy}{c^2},\qquad a,b,c > 0$$ My solution: The figure encloses area under itself so we're looking at: $$\left(\frac{x^2}{a^2}+...
programk5er's user avatar
0 votes
1 answer
25 views

Area of a cardioid and a circle

Given the Cardioid by $f(\varphi)=3-3\cos(\varphi)$ and the circle given by $g(\varphi)=-6\cos(\varphi)$. I have 2 questions regarding its areas: Why $\frac{1}{2}\int_{0}^{\pi}g(\varphi)^2d\varphi=9\...
MiguelCG's user avatar
  • 345
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0 answers
33 views

Area of ellipse using Green’s theorem [duplicate]

Given the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, which we want to calculate the area of. Parameterization $\mathbf{r}(t) = (a \cos t, b \sin t), \ t \in [0, 2\pi]$ My book says we can ...
math.lover's user avatar
1 vote
0 answers
64 views

Elegant derivation of area of an arbitraty 2D shape

Formula to derive Assume that we have an arbitrary nice 2D surface $S$ embedded in $\mathbb{R}^3$ parametrized as follows: $$ f\,:\,S\ni(\xi,\eta)\longmapsto(x(\xi,\eta)\,,\,y(\xi,\eta)\,,\,z(\xi,\eta)...
Jantur's user avatar
  • 76
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0 answers
28 views

What is the maximum area that can be enclosed in a polygon formed by n wires? [duplicate]

The problem is that we have n wires of different lengths, i.e. $w_1,w_2,...,w_n$. The wires are aligned in a way such that they enclose the maximum area. What is that maximum area, or its best ...
Panda's user avatar
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2 votes
0 answers
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Prove the area of a quadrilateral $A_1B_1C_1D_1$ has a local minimum at $A_1=A$

$ABCD$ is a cyclic quadrilateral. Its diagonals $AC,BD$ intersect at $P$. Let $E$ be the point on $AB$ such that $AE:EB=\tan\angle BAP:\tan\angle ABP$. Let $F$ be the point on $BC$ such that $BF:FC=\...
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  • 3,047
0 votes
0 answers
13 views

Hypothesis testing of Precision-Recall curve AUCs

In recent times, I have been about learning classification models (e.g., logistic regression) and how to evaluate them. While learning about the Precision-Recall (PR) curves, it occurred to me that ...
Yat-Hon's user avatar
  • 21
-1 votes
3 answers
92 views

Find the value of $\int_{-\infty}^{\infty} \cos^{-1}\left(\cos\left(\frac{24+4 x^2}{4+x^2}\right)\right)dx$. [closed]

Find the value of $\int_{-\infty}^{\infty} \cos^{-1}\left(\cos\left(\frac{24+4 x^2}{4+x^2}\right)\right)dx$. I tried separating the limits or converting it to a different trigonometric function. ...
bhargavi narayanan's user avatar
0 votes
1 answer
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How do I find the largest possible layout with the amount of money avaliable?

I have a question from a practice exam and I was wondering how to start this. Here's the Question: On a property next to a straight road, the owner is constructing a rectangular paddock. The side next ...
AlexanderWaller's user avatar
1 vote
1 answer
42 views

Find the area of a figure bounded by curves

Find the area of a figure bounded by curves $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,x^2+y^2=ab,x^2+y^2\geq ab,a>b$$ I called $x^2=ab-y^2$ and substituted the ellipse into the equation, which gave me ...
Dmitry's user avatar
  • 1,429
1 vote
2 answers
47 views

What is the reason for the ratios of square units not being the same as the ratios of units [closed]

When you have a square with side length 1 yard you get an area of 1 yd$^2$. When you convert the units of this square to feet you get a square with side length 3 and therefore an area of 9 square feet....
Sam's user avatar
  • 23
3 votes
4 answers
175 views

Area of the triangle inside the triangle

Area of each shape in the triangle is written. What is the area of the shaded region? Based on my search, $\dfrac{S_{\triangle MNP}}{S_{\triangle ABC}}$ can be calculated by Routh's Theorem. assuming ...
Soheil's user avatar
  • 6,794
1 vote
0 answers
45 views

How can this half-ellipse equation be integrated to find the surface area of the object?

How would the following equation be used to find an equation for the surface area of the 3d shape? $$y=0.7\sqrt{6.5^{2}-x^{2}}\left(0.05x+0.9\right)$$ The extension sheet described it as half of a ...
Gupert's user avatar
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1 vote
0 answers
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Area of Cyclic polygons

Can we generalize /extend Brahmagupta's formula to find the area of cyclic pentagons, hexagons, $n$ sided polygons $n>3$ ? For example $$ 2s= (a+b+c+d+e), \Delta=\sqrt{s (s-a)(s-b)(s-c)(s-d)(s-e)};...
Narasimham's user avatar
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5 votes
2 answers
174 views

Area of tight-angled $\triangle POB$ given extensions of $OP,BP$ to circle centred at $O$ through $B$?

We have a triangle $(\triangle POB)$ within a semicircle. $OP$ and $BP$ are extended to $OA$ and $BQ$. $AP = 5$ and $PQ = 7$. What is the area of the triangle? It's a problem I stumbled upon on ...
Afsar Ahmed's user avatar
3 votes
4 answers
807 views

A right-angled triangle has sides of integer length. Its area (in square metres) is twice its perimeter (in metres). What are the lengths of the sides

A right-angled triangle has sides of integer length. Its area (in square metres) is twice its perimeter (in metres). What are the lengths of the sides? The equations I have made so far is: Using ...
didlidoo's user avatar
1 vote
1 answer
56 views

Area of Cassini Oval

I am trying to find the area of the cassini oval, whose parametric and polar equations can be found here. To verify the result of area of Cassini Oval in the case $b>a$ written in Wolfram, I ...
Sam's user avatar
  • 3,360
1 vote
2 answers
74 views

What portion of the area of the parallelogram is the area of the figure bounded by these lines?

I encountered this problem: Each vertex of a parallelogram is connected with the midpoints of two opposite sides by straight lines. What portion of the area of the parallelogram is the area of the ...
pie's user avatar
  • 6,563
0 votes
1 answer
37 views

Knowing that the area of a lateral face is equal to the area of the base, find the measure of the angle formed by the planes $(MBN)$ and $(ABC)$.

the question Let $VABCD$ be a regular quadrilateral pyramid, $M$ the midpoint of the edge $VC$, $N$ the midpoint of the edge $AD$. Knowing that the area of a lateral face is equal to the area of the ...
IONELA BUCIU's user avatar
0 votes
1 answer
49 views

Finding the area between two parabolas with orthogonal axes.

I have two equations: $y=x^2+1$ and $x=(y-2)^2-1$. How would I find the area between these two curves? Should I rewrite one of the equations in terms of the other? For instance, rearranging $x=(y-2)^2-...
grxxes75's user avatar
2 votes
1 answer
53 views

Show that the sum of the perimeters of the circles is at most $\pi \sqrt{n}$ (the lines are not allowed to cut the interior of a subpolygon).

Question: Let us divide by straight lines a quadrangle of unit area into $n$ subpolygons and draw a circle into each subpolygon. Show that the sum of the perimeters of the circles is at most $\pi \...
Mods And Staff Are Not Fair's user avatar
8 votes
2 answers
246 views

What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?

Now asked on MO here. Given the length of the sides of a quadrilateral $a,b,c,d$ the area of the quadrilateral is less than or equal to $\frac{(a+b+c+d)^2}{16}$ i.e it is an upper bound of the area ...
pie's user avatar
  • 6,563
2 votes
4 answers
370 views

A parallelepiped has a volume of $216 \text{ cm}^3$ and the total area $216 \text{ cm}^2$. Prove that the parallelepiped is a cube.

The problem A parallelepiped has a volume of $216 \text{ cm}^3$ and the total area $216 \text{ cm}^2$. Prove that the parallelepiped is a cube. My question I don't understand if we are talking about a ...
IONELA BUCIU's user avatar
2 votes
1 answer
65 views

Can use one-variable integration to tell whether $R=\{(x,y):\;0\leq y\leq\left|\mathrm{sin}\frac{1}{x}|,\;0<x\ <1\right\}$ is rectifiable(has area)?

$R=\left\{(x,y):\;0\leq y\leq\left|\mathrm{sin}\frac{1}{x}\right|,\;0<x\ <1\right\}$, to determine whether it's rectifiable, can I use the method of one-variable integration to prove? Like this ...
Andrews's user avatar
  • 123
0 votes
0 answers
8 views

The maximum of $\sum_{k=0}^{n-1}a_{\sigma(k)}a_{\sigma(k+1)}$ for $\sigma\in S_n$.

I have interest of the area of radar chart for given data. The area depends on the order of data, so there is some order the area is maximum. I want to know the order. For mathematical argument, I ...
Yos's user avatar
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0 votes
1 answer
142 views

How do I solve for surface area in this case?

Okay, I have the parametric equation in spherical coordinates for a sphere, a cone tangent to that sphere and a circle inclined with an angle $\Omega$ to the $zy$ plane. ( Desmos graph link ). I need ...
Anonymous001's user avatar
1 vote
1 answer
55 views

Calculating the area enclosed by three graphs, but two out of them already enclose a different area.

I have three functions : $$y = 2x^2 + 2x \tag{1}$$ $$y = -x - 1 \tag{2}$$ $$x = 0 \tag{3}$$ Exercise asks for the area enclosed by three of those functions. It is clear that the area from $x = \frac{-...
Divine Orca's user avatar
0 votes
0 answers
34 views

Given the percent rain coverage in an area, what is the probability of hitting rain after driving X distance?

For simplicity's sake, I would say that the rain clouds are static. The question essentially becomes: given a finite area filled with random shaded regions covering X% of the area, what is the ...
Shaun Mitchell's user avatar
5 votes
0 answers
75 views

Geometrical meaning of calculating area using Green's theorem

Green's theorem says that: $$ \int_C L \ dx + \int_C M \ dy = \iint_D \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \ dx \ dy $$ If the M and N statisfy $\frac{\partial M}{\partial x} -...
Wojak2121's user avatar
7 votes
3 answers
290 views

Maximizing area of the triangle in a quarter circle

The radius of the quarter circle is $6\sqrt 5$ and we assume that $OA= 5$ and $OC=10$. What is the maximum area of the blue triangle? Interpreting the problem statement, I believe that points $A$ and ...
Soheil's user avatar
  • 6,794
1 vote
2 answers
109 views

Parametric area of a region bounded by two curves

Let $S(\epsilon)$ be the area of the region bounded by $y=e^x$ and $y=x+1+\epsilon$, where $\epsilon$ is a small positive number. When $\epsilon\to0,$ we have $$S(\epsilon)=S_0+\epsilon^\alpha S_1+\...
Math Student's user avatar
  • 5,352
0 votes
2 answers
56 views

Find loop passing through two points with length $L\pi$

Problem: Find a nice simple closed curve other than circle which passes through the points $(0,0)$ and $(1,0)$ on the Cartesian plane and whose length is $L\pi$. If the given condition is not the loop ...
Bob Dobbs's user avatar
  • 11.9k

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