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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9
votes
0answers
208 views

An amazing property of the Catenary

I discovered that if we want an arc of catenary in the interval $[a,b]$ we solve $$\int_a^b \sqrt{\cosh '(x)^2+1} \, dx=\int_a^b \cosh x \, dx$$ which means that the "result" of the length is equal ...
8
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1answer
124 views

Largest rectangle bounded under a function

Let $f$ be a positive monotonically increasing real function in $[0,1]$. Let $F$ be the area under the curve of $f$ ($F=\int_0^1{f(x)dx}$) For every $x\in[0,1]$, let $G(x)=f(x)\cdot (1-x)$ = the area ...
6
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0answers
67 views

Expected area of a random $n$-gon

Choose $n$ points $\{z_1, \ldots, z_n\}$ from the unit circle $\partial \mathbb{D} = \{z \in \mathbb{C}: |z| = 1\}$ uniformly at random, and let $P_n$ be the convex hull of the $z_i$'s. Let $X_n = ...
6
votes
1answer
150 views

Definite integral over a semicircular area $\int_0^{2a}\int_0^{\sqrt{2ax-x^2}}\frac{\phi'(y)(x^2+y^2)x}{\sqrt{4a^2x^2-(x^2+y^2)^2}}dy\,dx$.

change the order of integration in $$\int_0^{2a}\int_0^{\sqrt{2ax-x^2}}\frac{\phi'(y)(x^2+y^2)x}{\sqrt{4a^2x^2-(x^2+y^2)^2}}dy\,dx$$ I was able to change the order of integration here to $$\int_0^a\...
5
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0answers
71 views

Is this result already a known theorem in geometry?

I have been playing around with geometry and I found that: Let two perpendicular lines intersect at a point that is inside a circle. Then the area of the quadrilateral formed by the vertices made by ...
5
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0answers
402 views

How to calculate the area of the visible parts of a 3D PieChart?

I have created a 3D Pie Chart whose major feat (among the others) is to be rotated: I did it to demonstrate how the visual perception of data in a Pie Chart can be distorted depending on the position,...
4
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0answers
51 views

Find the maximum value of $\square OXPY$

Problem: There are moving point $X$ and $Y$ lie on the $x$ and $y$ axes, respectively. For moving point $P$, $PX=3$ and $PY=4$. Find the maximum area of $\square OXPY$.($O$ is origin). My solution: ...
4
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1answer
452 views

Did Euclid prove the formula for the area of a triangle?

In Proposition 6.23 of Euclid’s Elements, Euclid proves a result which in modern language says that the area of a parallelogram is equal to base times height. Now Euclid did not have the concept of ...
4
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0answers
131 views

What is the class of shapes with maximum area for a given volume and surface curvature?

If we consider a sphere with volume V and radius R, its surface area is minimal among all shapes of volume V. The radius of curvature of the surface is R at all points. What shapes will we obtain if ...
4
votes
1answer
107 views

The circumference of the circle is $C$, what is the area of circle in terms of $C$?

The circumference of the circle is $C$, what is the area of circle in terms of $C$? a). $\dfrac {C^2}{4\pi }$ b). $2\pi C$ c). $\dfrac {4}{3} \pi C^2$ d). $2\pi C^2$ My Attempt: $$\textrm {...
4
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1answer
86 views

Looking for some intuition behind why the area enclosed by a simple closed curve $C$ can be obtained by computing $\frac{1}{2i}\int_C {\bar{z}} \ dz$.

By some manipulation and an application of Green's Theorem, I am able to show that $$Area = \frac{1}{2i}\int_C {\bar{z}} \ dz $$ To me, this seems to be an unexpected result. Is there some intuition ...
4
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1answer
218 views

Area covered by Moving Circle?

Consider a situation where we have a point (x,y) moving on a 2-D plane. In fact, the point is function of time x=f(t),y=g(t). Centered around (x,y) is a circle of radius r? Obviously, we can visualize ...
4
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0answers
79 views

Area of convex hull of 6 points greater than twice the sum of the areas of 2 triangles formed by the 6 points

I noticed something interesting but I couldn't prove it to the end. I want to prove that if I have $6$ coplanar points and the area of convex hull of these points is equal to $P$, then I can mark the ...
4
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0answers
131 views

Equiareal Voronoi tessellation

I'm interested in even (or "proportional") disrtributions of points on 2D areas. Here is the initial question, but many others ideas appeared later. Centroidal Voronoi tessellation is well known ...
4
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0answers
829 views

Is there a formula for calculating the area of 2d shapes on a sphere?

Let's say I have 8 90° triangles on a sphere, like this, where all the angles are 90° when measured: I know that the area of one of those triangles will be (4πr2) * 1/8 as each triangle will take up ...
3
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1answer
82 views

How to calculate an area under $y=x^{-2}$ without integral

I need to get a formula for area under $y=x^{-2}$ for $x \in (1,a)$, where $a \in (1, +\infty)$, WITHOUT using integrals. I tried following: Let $h=\frac{a}{n}$, where $n$ is natural number of ...
3
votes
1answer
87 views

Maximum total area of n non-intersect circles?

Given n points on the x-axis, we give arbitrary radius for each point such that each constructed circle doesn't overlap another constructed circle from another point. Which means these circles do not ...
3
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2answers
495 views

Area of the region $\{(x,y):0\leq x \leq 1, 0 \leq y\leq 1, 3/4\leq x+y\leq 3/2\}$

Find the area of the region $\{(x,y):0\leq x \leq 1, 0 \leq y\leq 1, 3/4\leq x+y\leq 3/2\}$ (using definite integration). I cannot understand how to find this area. I have graphed the lines and found ...
3
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0answers
29 views

Computing hyper area of a contrained simplex

Let $D_n [r , (a_1, b_1 ) , \ldots , (a_n, b_n) ] = \{ (x_1 , \ldots , x_n ) \in \mathbb R^n \mid \sum_i x_i = r \mbox{ and } b_i \geq x_i \geq a_i \, \forall i \}$, where $r \geq b_i \geq a_i \geq 0$...
3
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0answers
48 views

Is there exists any general formula for shortest area of triangle with integeral sides where, perimeter is given?

if the perimeter of triangle is given , then, how to find the shortest area of triangle with integral sides. Let, $P$ be perimeter and $s$ be semi-perimeter. we known,$s=P/2$ area of triangle $=\...
3
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0answers
747 views

The area enclosed by curve $r^2=9\cos 5\theta$

Find the area enclosed ny the curve $r^2=9\cos 5\theta$. I have drawn its graph and it is the rose which has 5 petals. In order to find the total area it's sufficient to find the area of one petal. ...
3
votes
2answers
122 views

Help needed in understanding Heron's Formula

So i just started learning Trigonometry seriously and something doesn't feel right to me, either I'm missing something or not but. Lets assume we have a triangle and there are two ways to find the ...
3
votes
1answer
476 views

Tom Apostol - Calculus Vol. 1: Method of Exhaustion in Introduction

I'm new to proof based math, and I really want to get better from pursuing the discipline by myself. I actually had a fear, last year, of proof-based math after my first semester in college. It was ...
3
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0answers
55 views

What is the area of the Quadrilateral

This picture has no further description,from the place I saw. I tried to solve it using Menelaus's Theorem, and I came up to the equation: $$\frac{1}{CD} \times \frac{AF}{BF}=\frac{AE}{CE} \times \...
3
votes
2answers
157 views

Finding the area bounded by two curves

Find the area of the region bounded by the parabola $y = 4x^2$, the tangent line to this parabola at $(2, 16)$, and the $x$-axis. I found the tangent line to be $y=16x-16$ and set up the integral ...
3
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0answers
68 views

Variation of the Kakeya Needle Problem with positive thickness

What is the least area in the plane required to continuously rotate a needle of unit length and positive thickness $a \in ]0,1[$ around completely (i.e. $360°$)? The answer is ...
3
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0answers
6k views

Surface area of house

This is the image: So the question is: (a) Ralph is painting the barn, including the sides and roof. He wants to know how much paint to purchase. What is the total surface area that he is going to ...
3
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0answers
58 views

Fun Q2: Polygon inside a polygon. Find Test's area.

A triangle is drawn of side length $a$. Then a square is drawn inside the triangle such that the area of the square is maximum and the bottom side is shared. Then a regular pentagon is drawn inside ...
3
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0answers
218 views

Minimum number of circles to completely cover a sphere

I've encountered a problem which I have some idea of solving but am befuddled on how to proceed. Here is the full problem: Suppose that you wish to cover a 1-km radius, spherical planet with Wi-...
3
votes
1answer
73 views

Minimizing Area by Approximation

Suppose I have an increasing step function $E_c$ given by $$E_c(\phi) = \sum_{i=1}^n E_i \theta(\phi - \phi_i),$$ where $\theta$ is the Heaviside step function and $E_i$, $\phi$, and $\phi_i$ are all ...
3
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1answer
428 views

Need Help Finding Area of A Rectangle

I am really not sure if this is the right place to post a question like this, but I'm absolute stuck on this question. I would appreciate an answer greatly. A park is undergoing renovations to its ...
3
votes
1answer
92 views

Area of an equilateral triangle

Prove that if triangle $\triangle RST$ is equilateral, then the area of $\triangle RST$ is $\sqrt{\frac34}$ times the square of the length of a side. My thoughts: Let $s$ be the length of $RT$. Then ...
3
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0answers
647 views

Find the area and perimeter of a self-intersecting polygon using out of order coordinate points.

I was wondering if there was any algorithm or approach to find the perimeter and more importantly the area of a self-intersecting polygon using an array coordinate points. The problem is that ...
3
votes
1answer
152 views

Area enclosed between half lines in polar space

I don't know if the anwser to my question is obvious because I cannot find any explanation anywhere on google. Question The blue region $R$ is bounded by the curve C with equation $r^{2} = a^{2}cos(...
3
votes
2answers
2k views

Area of a triangle using vectors

I have to find the area of a triangle whose vertices have coordinates O$(0,0,0)$, A$(1,-5,-7)$ and B$(10,10,5)$ I thought that perhaps I should use the dot product to find the angle between the ...
3
votes
0answers
347 views

Upper bound for area of polygons

is there a formula for an upper bound for the area of a polygon, knowing the length of its edges? In the ideal situation, the answer would be a function $f_n(l_1,l_2,\dots,l_n)$ of the edges lengths (...
3
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1answer
402 views

Is my proof rigorous? (Archimedes area of parabola)

I am currently reading Apostol's Calculus volume 1 and was revising the part where the area of a parabolic segment is found. I decided to write my own proof similar to the one in the book, which I ...
2
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2answers
59 views

Three cevians in a triangle create four sub-triangles of area $1$. Find the area of a non-triangular region.

I'm having trouble proving that all the white and green areas have the same area, from there on we can obtain the answer $1+\sqrt5$ by proving that the inner red triangle points are midpoints.
2
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0answers
48 views

Surface area of melting ice block

A cubic block of ice is melting and retains its cubic shape as it melts. Its volume (in $\rm m^3$) at time $t$ is given by $$ V = 4000-2000 \cdot e^{0.01t}$$ As the ice melts, the bottom ...
2
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0answers
20 views

Find sub areas of a function in a circle

I have a cellular signal calculation function, which calculates the signal given the distance from the antenna. Without the constants, the function is basically: $f(d)=1/(d^α)$ where α is a parameter. ...
2
votes
0answers
33 views

Calculate the area bounded by $x^2+y^2=(\frac{x}{a})^3+(\frac{y}{b})^3$ and $x=0,y=0$

Calculate the area bounded by $x^2+y^2=(\frac{x}{a})^3+(\frac{y}{b})^3$ and $x=0,y=0$ We may assume $a,b>0$. Use the polar coordinate it's equivalent to value $\int_{0}^{\frac{\pi}{2}}\int_0^{\...
2
votes
0answers
57 views

What is the formula for the area between a curve and a parallel curve?

I recently took a liking to parallel curves and tried to find the area between them. Possible applications could be for making geometric swimming pools or some other area/volume based problem. The ...
2
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0answers
95 views

Bounded Area between $y=x^2$, and $y=9$, and $y=k$

I have been given a function to find the horizontal line $y=k$ which would splice the area bounded between $y=x^2$, and $y=9$, into two equal parts. I approached the function using symmetry to find ...
2
votes
0answers
125 views

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 ...
2
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0answers
153 views

What's the surface area of a Klein bottle?

I am creating a 3D model of a Klein Bottle based on the Robert Israel formula: Then I need to apply algorithms on the model and I need to know the surface area of this 3D model, then what's the ...
2
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0answers
50 views

What is Area of projection of curved circle?

We all know that ellipse is projection of a circle that lies in the vertical projecting plane. And what about projection of curved circle? Is it ellipse? Or another shape.
2
votes
0answers
75 views

Why is computing the perimeter of a circle with a circumscribed rectangle wrong, yet computing area under a curve with inscribed rectangles is valid?

There is a famous intuitive way to prove that $\pi$ is equal to $4$, by using square of perimeter $4$ with circle inscribed in it with radius of $1/2$. And now the question which arises to me is: What ...
2
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0answers
32 views

Extending Area Formulas to (Well-Behaved) Relations

According to my understanding, the formula to calculate the signed area between two functions (or axes, as when finding "area under a curve") $f(x)$ and $g(x)$ from $a$ to $e$ is $\int_a^e f(x) - g(x)\...
2
votes
0answers
40 views

Find area given two curves which belong to the orthogonal family to $x^2+2y^2=K$

Calculate the region area of the $xy$-plane limited by curves $C_1$ and $C_2$ with $|x|\leq4$, knowing that $C_1$ passes through $(1, 1)$, $C_2$ passes through $(1, -1)$ and both curves belong to the ...
2
votes
0answers
67 views

How to calculate part of sphere area bounded by two planes?

Let we have two planes $p_1$ and $p_2$ that intersect by an angle $\beta$. Let $p_1$ be on distance of $a$ from the center $C$ of sphere with radius $1$ and $a<1$. Let intersection of the two ...