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 ...
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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 ...
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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 ...
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1 vote
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $N$...
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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 <...
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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 ...
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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 ...
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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 ...
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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 ...
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