Area is a quantity that expresses the extent of a two-dimensional or three-dimensional surface or shape.

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Optimizing space for many shapes within an irregular shape

So let's say in a state, there are 50 schools dispersed throughout, given by Latitude Longitude points. How would we create distinct zones that optimize the space around each school? The goal is to ...
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83 views

Why prove that area is unique?

in the book Apostol's Calculus Volume 1, in the proof of the area of under of the parabola $x^2$ from $x=0$ to $x=b$ it is shown that the area $A$ must satisfy ...
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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 ...
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How do I calculate the area the Wigner Seitz cells cover in a square?

It's my first time here, so I appologise in advance if I break any rules through this post. So I have a Cartesian Lattice spanning across the Euclidean plane and a unit square. The lattice points ...
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Any idea how to approach this problem

A rectangular meadow will have a fence around it. The long side is $130$ m longer than the short side. The sides lengths can be written $x$ and $x+ 130$. Write a simplified expression for 1) ...
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25 views

Clarify formula that computes number of dies on wafer

I want to compute the number of dies per wafer (also DPW in the following). There are some formulas, that can be used to do so: ...
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2answers
54 views

Area of regular n-gon without trig?

As the title suggests I'm trying to find a formula for the area of a regular n-gon that doesn't use trigonometry. I already know the trig formula and I realize that my question is simply asking for ...
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2answers
52 views

given 3 circles, find relation of the regions

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. I had no idea how to find it nor where to start Note ...
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3answers
200 views

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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2answers
36 views

How to find the area under a semicircle using integration? [closed]

How would I go about finding the area under a semicircle? I know that to use integration the formula is $\int_a^b f(x) \mathrm{d}x.$However, when I put this into my graphing calculator it doesn't ...
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2answers
32 views

Find total area under infinite curves

My question is finding the total area covered by curves, such as the total area every curve in the following picture covers (from 100 on y axis to 200 on x axis): In my case, the curves are ...
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1answer
44 views

Triangular Area on hyperbolic surface

I have read numerous paper over area calculation in hyperbolic geometry but just can't seem to understand how to calculate a triangle's area in hyperbolic geometry. It would be nice to have a proof ...
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4answers
54 views

Find the area of the shaded region in the figure

Find the area of the shaded region in the figure What steps should I do? I tried following the steps listed here https://answers.yahoo.com/question/index?qid=20100305030526AAef8nZ But I got 150.7 ...
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6answers
562 views

Wanted : for more formulas to find the area of a triangle?

I know some formulas to find a triangle's area, like the ones below. Is there any reference containing most triangle area formulas? If you know more, please add them as an answer ...
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1answer
44 views

Line integrals - Surface area

Here is my task: Calculate surface area of $2(x^{2}+y^{2})^{2}=xy$ between surface $x^{2}+y^{2}=z$ and $z=0$. Here is my attempt to solve this problem. Firstly, I transformed line ...
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27 views

Geometric interperation of line integral - example

I have hard times figuring out geometric interpretation of line integrals. Here is one example from my book: Calculate area of cylinder $x^{2}+y^{2}=ax$ sliced with sphere $x^{2}+y^{2}+z^{2}=a^{2}$. ...
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7 views

Need help with getting 95% ellipse area of a 2D PSR plot [closed]

So I have been referring to the method of obtaining this as outlined in the following paper (pg 4 - 5): ...
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2answers
270 views

Area of irregular hexagon with all angles = 120 degrees

I want to derive a formula to calculate the area of a irregular hexagon which is guaranteed to have all internal angles = 120 degrees. Please guide me how to proceed to form a general formula.
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3answers
40 views

Calculate pentagon area based on lengths of all its sides

Sorry for this question. I guessed there is an online calculator to calculate the area of the pentagon if we know lengths of all its five sides. So, here are the lengths of sides of pentagon ABCDE: ...
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1answer
32 views

Alternative proof of lateral surface area of a conical frustum

I am trying to come up with an alternative proof of the lateral surface area of a conical frustum with parallel bases by making use of the linear increase in perimeter $P$ of the base with respect to ...
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3answers
137 views

What is the area in which the two goats can eat grass, if they choose not to eat in the common approachable area?

Two goats are tied with a rope of length 40m outside of a rectangular shed of dimensions 50m X 30m. The goats are tied to different corners which lie on the opposite ends of a diagonal of the shed. ...
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1answer
55 views

Area form and surface area

I know how one can define the surface area via the charts of a surface in $\mathbb{R}^3.$ click here for instance Now, I read that the canonical surface area form for such a surface with surface ...
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35 views

What's the area of the shape defined by all points whose distances from two focal points multiply to give the same product?

This shape, which I call the multiplicoid, is the equivalent of, and very similar to, an ellipse. However, instead of the distance between each point and the two focal points summing to a constant, ...
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39 views

What is the area of the shape defined by the locus of a point on a circle rolling around another circle?

What is the area of a shape, which I'm deeming a 'cylicoid', which is defined as follows: Circle A of radius 1 is held stationary. Circle B of radius 1 has a point on its rim which traces a path as it ...
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25 views

Area of equilateral triangle from circumcircle

I am trying to calculate skewness of triangle. Given the sides of a triangle (not equilateral), I calculated circumradius from which I would like to get area of equilateral triangle.
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56 views

Area of the figure within the circle and outside a polygon

For which values of the parameter $c \in \mathbb{R}$, the area $S$ of the figure $F$, consisting of the points $(x,y)$ such that $$\begin{gathered} \max \{ \left| x \right|,y\} \geqslant 2c \hfill ...
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70 views

Why can't the nth triangular number be expressed as the area of an equilateral triangle?

It should be self-intuitive that the $nth$ triangular number is an equilateral triangle with base $n$, and thus its area should equal the value of the triangular number. So, I was wondering: why ...
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1answer
27 views

Combined overlap area question

I'm having a bit of trouble with this question: Sue works in a rug shop. She as three rugs with a combined area of $200 m^2$. She arranged the rugs so they overlap and cover a floor area of $140 ...
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1answer
40 views

Surface area of $y = \sin(\pi x)$, from $x=0$ to $2$, rotated about the $x$-axis.

When I use the surface area formula I get 0, and Wolfram got zero as well when I use the bounds 0 to 2, why is this? However the solution manual uses the integral with bounds 0 to 1.. What is going ...
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2answers
48 views

Calculate area of “hand-drawn” polygon

I have a series of coordinates that represent a hand-drawn polygon. At the intersection, the lines slightly "overshoot," e.g.: ...
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3answers
65 views

Find the area of triangle, given an angle and the length of the segments cut by the projection of the incenter on the opposite side.

In a triangle $ABC$, one of the angles (say $\widehat{C}$) equals $60^\circ$. Given that the incircle touches the opposite side ($AB$) in a point that splits it in two segments having length $a$ ...
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2answers
26 views

Cylinder volume with curved base area

Suppose, we have a cylinder with a flat base area $A$ and height $H$. The volume $V$ of the cylinder is obtained by multiplying the two quantities: $V=AH$. But what happens, when the base surface is ...
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1answer
53 views

Area of overlapping squares

I'm working on a programming project and got to the point where I need to find how much is the blue square overlapping each of the other 9 squares. The squares' sides(including the blue one's) are ...
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Area of region - double integral

Here is my task: Calculate area of region $(x^{2}+y^{2})^{2}\leq a^{2}(x^{2}-y^{2})$. Here is what I have done. After transforming this line to polar form $(x=\rho\cos\phi,y=\rho\sin\phi)$, we have: ...
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Area of region - Double integrals

Here is my task: Calculate area of region $(\frac{x}{a}+\frac{y}{b})^{5}=\frac{x^{2}y^{2}}{c^{4}}$,$a,b,c>0$. Solution is $A=\frac{a^{5}b^{5}}{1260c^{8}}$ Any idea how to solve this?
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Co-ordinate geometry and area of triangle

When a straight line $ax+by+c=0$ forms a triangle with the axes $x$ and $y$, what is the general formula for the area of the triangle?
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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 ...
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1answer
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How many square millimetres are in 0.000075 square metres?

Doing my head in a bit.. $0.000 075 \times 100$ = in square cm, and times that by $10 =$ square mm but apparently I'm off by a couple of orders of magnitude.
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1answer
61 views

Find the area bounded between $f(x)=\frac{\arctan(x)}{x^2}$ and $g(x)=\frac{\arctan(x)}{x^2+1}$

Find the area bounded between $$f(x)=\frac{\arctan(x)}{x^2} \quad\text{and}\quad g(x)=\frac{\arctan(x)}{x^2+1}.$$ The title says the question. The limits are from 1 to infinity. I know that I ...
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1answer
58 views

How to calculate the area of a region with a closed plane curve boundary?

Under the conditions of Green’s Theorem, the area of a region $R$ enclosed by a curve $C$ is $$\oint_C x \, dy=-\oint_C y \, dx=\frac{1}{2}\oint_C (x \, dy - y \, dx)$$ I tried to use the result to ...
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1answer
52 views

Minimal area of triangle

We have the points $A(2, 3-m), B(m+2, -1)$ and $C(m, 2-m)$. Where $m$ is a real number. Find $m$ for which the area of triangle $ABC$ is minimal. So I've tried to find the equation of line $BC$(the ...
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2answers
56 views

How to calculate the area of a parabolic dish

I feel like this should be really easy, but I'm not sure if I'm doing it correctly so I'm going to give it a go here, and if I'm not very good at maths (I'm not) then you can hopefully correct me! ...
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How is the area of a circle calculated using basic mathematics?

Area of a circle is addition of circumference of layers of a onion. If n is radius of a onion then area is $$ A = 2 \pi \cdot 1 + 2 \pi \cdot 2 + 2\pi \cdot 3 + \ldots + 2 \pi \cdot n $$ which $$ ...
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Circles in circle

If we are given one big circle and infinite amount of smaller circles with equal radius (of course radius of the smaller is < radius of the big one) and we have to put in the center of the big ...
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Uniqueness of a number $area(A)$

I have the following definition of area: Let $A$ be a bounded set from $\mathbb{R}^2$. We say that $A$ has area if there exist two sequences $(E_n)_{n\in \mathbb{N}}, (F_n)_{n\in \mathbb{N}}$ of ...
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Finding area by integration, increasing inaccuracies with complex functions?

I am looking for an explanation as to why the method of integration to find the area of function using limits provides a greater % difference between other methods (In this example Simpsons) with ...
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1answer
57 views

Finding the area between $x \sqrt{4x-x^2}$ and $\sqrt{4x-x^2}$

So I've been doing real analysis for a last couple of days, and stumbled upon this task. The task is to find the area enclosed by $$y_1=x\sqrt{4x-x^2} $$ and $$y_2= \sqrt{4x-x^2} $$ This is one of ...
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2answers
110 views

Surface area of the ellipsoid $\frac{x^2}{16}+\frac{y^2}{8}+z^2=1$

My professor gave us this question on a calculus II quiz. One of my calculus III pals suggested I use surface integrals, but that tool is not available to us (I don't know how to use it yet, nor do my ...
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0answers
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Help with Apostol's “Calculus, vol. 1”, Section 1.18

In section 1.18 ("The area of an ordinate set expressed as an integral"), Apostol proves two theorems. the first, theorem 1.10, deals with the area of a function's ordinate set; the second, theorem ...
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1answer
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Straight Lines; The area enclosed by |x| +|y| =1 [closed]

Find the area enclosed by the following graph : $|x| +|y|=1 $