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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168 views

If $\text{Area} (A) = \text{Area} (B)$ and $\text{Perimeter}(A) = \text{Perimeter}(B) \implies A \cong B$?

If I have an $n$-gon $A$ and a convex $n$-gon $B$ with the same perimeter and the same area, does $A\cong B$? Edit : What becomes the answer if I replace convex by regular?
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Are of quadrilateral: $S \leq \frac{(a+b)(c+d)}4 $

I got stuck on this problem: Given a convex quadrilateral of area $S$ and sides $a$, $b$, $c$ and $d$, prove that: $$S \leq \frac{(a+b)(c+d)}4$$ What I've done so far was to proof that $$S ...
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Alternative area of a triangle formula

The problem is as follows: There is a triangle $ABC$ and I need to show that it's area is: $$\frac{1}{2} c^2 \frac{\sin A \sin B}{\sin (A+B)}$$ Since there is a half in front I decided that base*...
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Expected value of area of triangle

Here is the problem: Let $A$ be the point with coordinates $(1, 0)$ in $\mathbb R ^2$. Another point $B$ is chosen randomly over the unit circle. What is then the expected value of the area of the ...
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Area of the field that the cow can graze.

How do we find the area that the cow can graze? The question goes as follows-- There is a circular barn house surrounded by a huge grazing field. A cow is tied to the rope ($AB$) at the end $A$ as ...
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Finding the area under a curve represented by the equations $x=a\cos{t}+\frac{a}{2}\ln{\left(\tan^2{\frac{t}{2}}\right)}$ and $y=a\sin t$

How do I find the area of the curve represented by the following equations, $$x=a\cos{t}+\frac{a}{2}\ln{\left(\tan^2{\frac{t}{2}}\right)}\\ y=a\sin t$$ Here's what I tried: Let $A$ denote the area ...
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Area between two polar curves $r = 2 \sin\theta$ and $r =2\cos\theta$

I am trying to find the area between the polar curves $r = 2 \sin θ$ and $r = 2 \cos θ$. I set up the area equation as follows: $$\frac12\int_0^{\pi/4}((2\sinθ)^2-(2\cosθ)^2)\,d\theta$$ I could not ...
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Finding the area enclosed by curve defined by $\arcsin x+\arcsin y=\arcsin(x\sqrt{1-y^2}+y\sqrt{1-x^2})$

If $\arcsin x+\arcsin y=\arcsin(x\sqrt{1-y^2}+y\sqrt{1-x^2})$ Then the area represented by the locus of point $(x,y)$ if it is given that $|x|,|y|\leq 1$ My Try: Put $x=\sin \alpha$ and $y=...
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Integration - Area under a curve

I was working on a computer programming project that involves 2D drawing on windows OS. I was displaying curves using simple mathematical formulas, and was thinking of filling the part under a curve. ...
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Need help with level 3 Calculus problem

Two curves $$C_1: ([f(y)]^{2/3} + ([f(x)]^{1/3}) = 0\quad\mbox{and}\quad C_2: [f(y)]^{2/3}+ [f(x)]^{2/3} = 12, $$ satisfying the relation $$ (x-y)f(x+y)-(x+y)f(x-y) = 4xy(x^2-y^2).$$ 1.) Evaluate ...
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Can I square the triangle?

I know I can't construct a square with the same area as a given circle (because $\pi$ is transcendental). Can I construct (ruler and compass) a square with the same area as a given triangle? I ...
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Why is the volume one third of that? I mean, where's the fault in my logic? [duplicate]

The volume of a cuboid is $l \times b \times h$. That is, it is equal to base area times height. I think it means that the base is added up height times, it becomes volume (It makes sense if we think ...
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Inequality in triangle involving side lenghs, medians and area

A, B and C are the vertices of a triangle. Denote $m_a$, $m_b$ and $m_c$ the medians from A, B and C. Prove the inequality: $$\sum_{cyc}{a^2bcm_a}\geq\sum_{cyc}{cS(a^2+b^2)}$$where a, b and c are the ...
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The Area of an Irregular Hexagon

So I have a hexagon with $3$ side lengths of length $2$, and $3$ side lengths of length $1$. All side length of length $1$ are next to each other, and all sides of length $2$ are as well. The two ...
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Calculating the area between two functions expressed in polar coördinates

I have the following to polar coördinates: $r=1+\cos(\theta)$ and $r=3\cos(\theta)$. The question is to calculate the area in side $r=1+\cos(\theta)$ and outside $r=3\cos(\theta)$. I know I need to ...
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Finding Angle using Geometry

In an equilateral triangle $ ABC $ the point $ D $ and $ E $ are on sides $ AC $ and $AB$ respectively, such that $ BD $ and $ CE $ intersect at $P$ , and the area of the quadrilateral $ ADPE $ is ...
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How to calculate area of triangle via summation of internal rectangles?

For constant velocity, one could directly prove area of a rectangle whose length is time traveled and height the velocity, is actually the displacement. I would like to extend the same area logic ...
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Find the area of curve

The folium of descartes is defined by $x^3+y^3-3xy=0$ (cartesian), $r=\frac{3\sec \theta \tan \theta}{1+\tan^3 \theta}$ (polar) or $x=\frac{3at}{1+t^3}, y=\frac{3at^2}{1+t^3}$ (parametric). It looks ...
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Understanding Leibniz's Formula for $\pi/4$, geometric proof

I was reading about Leibniz's geometric proof for $\pi/4$ in this wiki-page. I understand this proof almost completely with the exception of one part where it is stated that: $$dC=\bigtriangleup ...
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Related Rates Ladder Question

A ladder 25 feet in length creates a right triangle with the wall it's leaning against. If the base of the ladder is being pulled horizontally away from the wall at a rate of 2ft/s, what is the rate ...
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Spherical Triangle

I know that the area for a spherical triangle is calculated as Area $= r^2(a+b+c-\pi)=r^2E$ where $E= (a+b+c-\pi)$ is the spherical excess I was wondering why do you have to multiply by $r^2$ (the ...
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Measure of a set $A=\bigcup_{z_x,z_y} \{ (x,y) \in [0,1]^2 : |axz_x-byz_y| \le c \}$

What is the measure of the set $$ A=\bigcup_{\substack{1 \le z_x \le N_x \\ 1 \le z_y \le N_y}} A(z_x,z_y) $$ where $A(z_x,z_y)=\{ (x,y) \in [0,1]^2 : \lvert axz_x-byz_y \rvert \le c \}$ for ...
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Expressing the area as a function :)

Express the area A of an equilateral triangle as a function of the height of the triangle. Thanks :) I am not sure where to even start on how to answer this problem.
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2 calculus questions with integration - check me

I have 2 questions I would like assistance with. 1) Find the area of the region bounded by the graphs $y=5x, y=15x, y=\frac{4}{x}, y=\frac{8}{x}$ This was very difficult and tedious. I had trouble ...
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Area in Polar Coordinates

I have tried to solve this problem by subtracting the area of the whole by the smaller area. I got up to $$\int_{1/2}^{11\pi/6} 12 (\cos (\theta)-6)^2 \mathrm{d}\theta- \frac12 \int_{-\pi/6}^{\pi/6} ...
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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 ...
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Finding area problem

There is this simple geometry question that seems so easy but I think the question lacks some information (does it?). Or maybe there are other ways to solve the problem. So the problem says, there ...
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129 views

$I = \int_{- \infty}^{\infty} \delta (n - ||\mathbf{x}||^2) \mathrm d \mathbf{x} $ should not diverge

I'm having trouble evaluating this integral: $$I = \int_{-\infty}^\infty \delta (n - ||\mathbf{x}||^2) \mathrm d \mathbf{x} $$ where $\delta(x)$ is Dirac delta function, $\mathbf x$ is the real $n$-...
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In a square ABCD with side 14cm, 2 quadrants were made with centres A & B respectively and AB as radius. Find area of region I and II

https://photos.app.goo.gl/5ibXDN5u6s0yo6KB3 I could do the following: II + III = $\frac{1}{4}$ × $π$ × $14^2$ = $49π$ = $154cm^2$ II + IV = $\frac{1}{4}$ × $π$ × $14^2$ = $49π = 154 cm^2$ Area of ...
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Exact Answer: Cut a circular pizza into 16 Pieces, 12 of them with equal area. Find where the cuts should be made to satisfy these conditions.

EDIT: It's almost been a year since I attempted this problem, so I decided to come back to it. One thing I was trying to do before was find a "closed form" of this problem, but really I wanted a way ...
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Area of intersection of polynomial inequalities.

Fix a point $(p_1, p_2) \in (0,1)^2$ and let $\epsilon \in (0,1)$. Consider now the following region \begin{equation} \mathcal{D} = \{(x,y)\ :\ (x,y) \in (0,1)^2,\ n_1 x^i + n_2 y^i - (n_1 p_1^i +...
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Existence of area for a norm sphere

Let $\lVert \cdot \rVert \colon \mathbb{R}^d \to \mathbb{R}^{\geq0}$ be a norm. Prove that the area $A(S)$ of the unit-sphere $S = \{x \in \mathbb{R}^d : \lVert x \rVert = 1\}$ exists. Integral ...
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Find area of rectangle

This is not kind of homeworks and please teke easy to consider. I found interesting problem which is very elementary but not easy (for me). The problem is to find an area of this rectangle. I have ...
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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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Areas between intersecting chords

In the circle below let the two chords be called $C_1$ and $C_2$, and their intersection be some point that is not the center. The chord power theorem tell us that $a \cdot b = c \cdot d$. I am ...
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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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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: ...
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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 ...
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Is there a simple proof for the (co)area formula in the case of equal dimensions?

Let $\Omega \subseteq \mathbb{R}^n$ be a nice domain (say a ball), and let $f:\Omega \to \mathbb{R}^n$ be smooth. Then $$ \int_{\Omega} |\det df|=\int_{\mathbb{R}^n} |f^{-1}(y)|dy, \tag{1}$$ where $|f^...
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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 ...
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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 {...
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$ABCD$ is a quadrilateral and $P,Q$ are midpoints of $CD, AB.$…

$ABCD$ is a quadrilateral and $P,Q$ are midpoints of $CD, AB.$ $AP$ and $DQ$ meet at $X, BP$ and $CQ$ meet at $Y.$ Prove that $$|ADX|+|BCY|=|PXQY|$$ (here $|N|$ means area of the shape $N$) I have ...
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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 ...
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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 ...
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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 ...
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Area of triangle given 3 equations of sides, without finding points of intersection [duplicate]

Can anyone tell me a way to calculate area of a triangle given the equations of its 3 sides without sketching or finding the points of intersection?
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319 views

Area of a square inscribed in a circle

ABCD is a square inscribed in a circle whose diameter is L cm. If P and Q are mid points of BC and CD, respectively, find the shaded area MDCNT Thanks I tried this If I knew the M value I could ...
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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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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 ...
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Finding the area of the shaded square inside a square created by connecting point-opposite midpoint [closed]

If the lines intersect the vertices of the square. The area of the square is $1$ and the lines also intersect the midpoints of the square lines. How to find the area of shaded region?