Questions tagged [area]

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

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Are closed simple curves with that property necessarily circles?

This is a more interesting follow-up to the question Are closed simple curves with this property necessarily circles? Let $\gamma:[0,1]\to \mathbb R^2 $ be a closed simple $C^1$ convex curve and $\...
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Triangles area question

This question came in RMO, an olympiad in India. I solved it but with the assumption that the lines are parallel, though we are not given this info in the question. In acute $\triangle ABC$, let D ...
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Find the area lying inside the cardioid $r=1+\cos\theta$ and outside the parabola $r(1+ \cos\theta)=1$

I need to find the area lying inside the cardioid $r=1+ \cos\theta$ and outside the parabola $r(1+ \cos\theta)=1$. ATTEMPT First I found the intersection point of two curves which comes out to be $\...
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Area of a surface using integration. Confusion with aspect of formal definition.

My textbook explains that, when finding the area of a surface using integration, we approximate each surface element by $$\left| \Delta u \dfrac{\partial \overrightarrow{r}}{\partial u} \times \Delta ...
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1answer
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additive integral property

There's a common property of definite integrals: $\int_a^bf(x) \, dx=\int_a^cf(x)\,dx+\int_c^bf(x)\,dx$. I've often seen it said that $c$ must lie in the interval $[a,b]$. However, is this really the ...
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Problems with ln(ax) equations.

After fiddling around with the ln() function, I arrived at a problem. I have found that $a \approx 1.39095$. However, I couldn't find the exact value. Using the Lambert w function, I have already ...
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Calculate the viewing-angle on a square (3d-calc)

I'm in big trouble: My program (Java) successfully recognised a square drawn on a paper (by its 4 edges). Now I need to calculate, under which angle the webcam is facing this square. So I get the 4 ...
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A question related to triangles , areas , ratio of areas of triangles.

I know the title is confusing but that is because of 150-character limit, if anyone of you can improve it , please do. Consider $\triangle ABC.$ Choose a point $D$ on segment $BC$ such that $BD/DC=1/...
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Finding the area of an ellipse using Divergence Theorem

As an example of Divergence Theorem, our textbook mentions finding area of an ellipse, but it isn't clear how it was derived though. Following is an excerpt from the textbook. Suppose there is an ...
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Is this a valid way of deriving the area of a circle?

On the Wikipedia article about deriving the area of a circle, it mentions that the formula $$ \text{area} = \pi r^2 $$ can be derived by evaluating the integral $$ 2 \int_{-r}^{r} \sqrt{r^2-x^2} \, dx ...
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Least triangular convex polygon

(This question is based on a question posed in a math riddle post on Reddit.) Let $P$ be a convex polygon. Let the non-triangularity of $P$ be the minimum area of the symmetric difference (shown with ...
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Generalization of two formulae and an alternative proof of Bretschneider's formula

Below I present two seemingly unknown identities that I then use to provide an alternative proof of Bretschneider's formula. Made the necessary adjustments, the identities can also be used to provide ...
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Filling a square with unequally sized circles

Suppose we are given a set of circles with integer radii 1, 2, 3 ... $n$. What is the smallest square which they can all fit in such that they do not overlap? For instance, when $n$=1, clearly we can ...
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Area inside $|x|^G+|y|^G=r^G$ is an integer multiple of $r^2$

The problem is to find the numbers $G_i$ such that the area inside the curve $|x|^{G_i}+|y|^{G_i}=r^{G_i}$ is an integer multiple of $r^2$. Because the curve defined by this equation is always ...
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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,...
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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 ...
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Area of Intersection of Circle and Square

Given a point $(x,y)\in [0,1]^2$ and $r > 0$, I would like to derive a general formula for the area of the intersection of the circle of radius $r$ centered at $(x,y)$ and the unit square. What is ...
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Find the length and width of rectangle when you are given the area

The area of a rectangle is $x^2 + 4x - 12$. What is the length and width of the rectangle? The solution says the main idea is to factor $x^2 + 4x -12$. So, since $-12 = -2 \times 6$ and $-2 + 6 = 4$...
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Area of a parallelogram, vertices $(-1,-1), (4,1), (5,3), (10,5)$.

I need to find the area of a parallelogram with vertices $(-1,-1), (4,1), (5,3), (10,5)$. If I denote $A=(-1,-1)$, $B=(4,1)$, $C=(5,3)$, $D=(10,5)$, then I see that $\overrightarrow{AB}=(5,2)=\...
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Area of ellipse which is not in standard form

By graphing device i understand that $x^2+xy+y^2=1$ is ellipse. By some geometry i find area of above ellipse which comes out $\pi$ (is it right?), but it was easy case. Is there any quick method or ...
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A geometry question…

In the given figure, $ABCD$ is a square of side $3$cm. If $BEMN$ is another square of side $5$cm & $BCE$ is a triangle right angled at $C$. Then the length of $CN$ will be:- I plotted this on ...
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Prove that $1<\int_{0}^{\frac{\pi}{2}}\sqrt{\sin x}dx<\sqrt{\frac{\pi}{2}}$using integration.

Prove that $$1<\int_{0}^{\frac{\pi}{2}}\sqrt{\sin x}dx<\sqrt{\frac{\pi}{2}}$$ using integration. My Attempt I tried using the Jordan's inequality $$\frac{2}{\pi}x\leq\sin x<1$$ Taking square ...
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Calculating area of two vectors. (problems with getting calculations correct)

We have two vectors $u$ and $v$: $$ u=-3i+5j+2k $$ $$ v=4i+3j-3k $$ Cross product $u\times v$ gives us: $$u \times v =\begin{vmatrix}i & j & k \\ -3 & 5 & 2 \\ 4 & 3 & -3 \...
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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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Area bounded with curves [closed]

I need to calculate area bounded with curves $(x^2+y^2)^2\le 3(x^2-y^2), x^2+y^2\le \sqrt{2}x$. Here we have inside of leminscate and circle. I think i should use that $P=\iint dxdx$ but not sure how.
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What is the area of the triangle ABC?

$ABC$ is an equaliteral triangle. Suppose $DB=4$, $DA=4\sqrt{3}$ and $DC=8$. Find the area of the triangle $ABC$.
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How to find top and bottom function for finding area between two curves?

When I am trying to find the area between two curves below the $\mathbf{x}$-axis, should I always have the function furthest away from the x-axis as the function you are subtracting from, or should it ...
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Integral Calculus, Infinitesimal

To integrate $y=f(x)$ from $a$ to $b$ we break the function into small rectangles of width $dx$. So the $n$-th rectangle will be at a distance of $n\,dx$ from $a$ on the $x$-axis. Let there be $t$ ...
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Compute the area of a given set

Problem I have the set $D$ defined as: $$ D=\{(x,y)|x\ge 0, 0 \le y \le 64-81x^2 \} $$ I want to compute the area of this set $A(D)$ Attempt to solve You could try to solve the area of this set ...
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Area under quarter circle by integration

How would one go about finding out the area under a quarter circle by integrating. The quarter circle's radius is r and the whole circle's center is positioned at the origin of the coordinates. (The ...
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Integrals and area of circle

I have no idea where to start this one guys: Write a definite integral whose value is the area of the region between the two circles: $$x^2+y^2=1$$ $$(x-1)^2+y^2=1$$ Do I need to find my ...
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2answers
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How prove this $S_{\Delta ABC}\ge\frac{3\sqrt{3}}{4\pi}$

There is convex body $T$ (with the area is $1$), show that there is a triangle $\Delta ABC$, such $A,B,C\in T$, and $$S_{\Delta ABC}\ge\dfrac{3\sqrt{3}}{4\pi}$$ This problem is from China The ...
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Find all triangles of which perimeter and area are numerically equal

Find all triangles of which perimeter and area are numerically equal. I have got solution for right angle triangles but not of others
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The maximum area of a hexagon that can inscribed in an ellipse

I have to find the maximum area of a hexagon that can inscribed in an ellipse. The ellipse has been given as $\frac{x^2}{16}+\frac{y^2}{9}=1$. I considered the circle with the major axis of the ...
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3answers
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Triangle area inequalities; semiperimeter

I've got stuck on this problem : Proof that for every triangle of sides $a$, $b$ and $c$ and area $S$, the following inequalities are true : $4S \le a^2 + b^2$ $4S \le b^2 + c^2$ $4S \le a^2 + c^2$ $...
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Finding the area of a square that has a circle inside itself

I tried to solve the following problem: I think the image is self-descriptive. I tried to draw a vertical line from the top-end of $\theta$ angle to the horizontal line, then tried to use the ...
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Optimization with cylinder

I have no idea how to do this problem at all. A cylindrical can without a top is made to contain V cm^3 of liquid. Find the dimensions that will minimize the cost of the metal to make the can. Since ...
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3answers
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Area bounded by $2 \leq|x+3 y|+|x-y| \leq 4$

Find the area of the region bounded by $$2 \leq|x+3 y|+|x-y| \leq 4$$ I tried taking four cases which are: $$x+3y \geq 0, x-y \geq 0$$ $$x+3y \geq 0, x-y \leq 0$$ $$x+3y \leq 0, x-y \geq 0$$ $$x+3y \...
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Why is $f(4)$ the area under $f'(x)$ specifically from $0$ to $4$ and not for ex from $1$ to $4$ or $2$ to $4$?

I've seen the geometric argument for why any differentiable function $f(x)$ gives the rate of change of the area under its own curve to $x$ for a specific input $x$, and it makes sense to me. It also ...
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1answer
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Is the matrix filled with the areas of pairwise intersections of disks in a plane always positive semidefinite?

Consider disks $s_1, \cdots, s_n$ in the plane and let $a_{ij}$ be the area of $s_i\cap s_j$. Is it true that for any real numbers $x_1,\cdots, x_n$ we have $$ \sum_{i,j=1}^n x_ix_j a_{ij} \geq 0$$ ...
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Maximum area of a fenced playpen on the side of a house.

Here's an interesting problem: you just got a really cute puppy, and you want it to have a large rectangular playpen to run around in. What's more, your neighbor just happened to have 100 feet of ...
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For a point K inside a triangle show an equality

Let ABC be a triangle whose heights are $h_a,h_b$ and $h_c$. Let $K$ be any point inside the triangle, and $d_a,d_b$ and $d_c$ the distances of $K$ from the sides $a,b$ and $c$, respectively. Show ...
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Converting cartesian to polar double integral

Convert the following integral to polar coordinates. You do not need to evaluate. $$\int_{-3}^3 \int_{x}^{\sqrt{9-x^2}} x^2y dy dx$$ My work : I plotted the limits and I don't understand the bounded ...
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Calculating area of a shape with circular boundaries with elementary methods

Question. $\square ABCD$ is a square with $AB = 10$. Circle $O$ inscribes the $\square ABCD$. The center of the arc is $A$. What is the area of the colored area? Explanation: This problem can be ...
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Calculating the area with simple and double integrals

I have an exam on calculus next week and I'm confused with the usage of simple and double integrals. As far as I know, if you want to calculate the area under the curve of a function $f(x)$ between $a$...
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1answer
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Dividing a square into two regions with minimal interface

We want to divide a unit square into a black region of given area $A\in[0,1]$ and a white region of area $1-A$, while minimizing the interface perimeter between the two regions. Let's say both regions ...
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1answer
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Moment of inertia for a n sided regular polygon

It is as simple as the title. We have an area $A_n$ of an homogeneous n-sided polygon with the density of $\rho$. With its center in $(0,0)$, and one of it's points in $(a,0)$, $a>0$. Calculate $...
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What is the area bounded by the curve $r^2 + \theta^2 = 1$?

What is the area bounded by the curve $r^2 + \theta^2 = 1$? (given in polar coordinates) My approach was to calculate the definite integral: $$\frac12 \int_0^1 (1-\theta^2) \, d\theta$$ Integration ...
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Possible alternative for finding the Area under the floor function (aka, the integral of floor(x))

So, I had to ask myself the question as to what the area under the floor function could possibly be. I started by graphing the basic $\mbox{floor}(x)$ function (I personally use desmos.com for a nice ...
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Area of the part of a sphere

Let $x \in \mathbb{R^3}$ and $t>0$. Also let $R>0$. Why is the area of the part of the sphere $\partial B(x,t)$ inside $B(0,R)$ smaller than the area of $\partial B(0,R)$: $$|\partial B(x,t) \...

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