# 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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### How can i Prove that the gray area is the same as white area? [duplicate]

A circle is cut into 8 parts, each part has the angle 45 degrees from an arbitrary point. how to prove that the white area is the same as the Gray area? I just want any hint for solving this question....
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### Area bound by $y =\ln x +\tan^{-1} x$?

I'm not really sure how to do this without having a real graph using a graphing calculator. Adding the integrals would no work since some areas are overlapping, and even subtracting doesn't since one ...
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### Finding $\iiint 6z\,dx\,dy\,dz$ over $\lbrace (x,y,z) \in \mathbb{R}^3 : |x+y| \le z \le |x|+|y|\le 1 \rbrace$

I want to calculate integral : $\iiint 6z\,dx\,dy\,dz$. The area is $$\Omega= \lbrace (x,y,z) \in \mathbb{R}^3 : |x+y| \le z \le |x|+|y|\le 1 \rbrace$$ My problem is this, that I don't know what is ...
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### Determine the area enclosed by the curve

Determine the area enclosed by the curve with two polar equations: $r= \sqrt 2 \sin(\alpha)$ $r^2 = \sin(2\alpha)$ I have no clue how to do this. A formula we are given is the one below but I'm ...
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### Number of Atoms on Earth's Surface (Question based on Vsauce video which involves fractal dimension)

In a Vsauce video titled "How much of the Earth can you see at once" they try to do a calculation to estimate the number of atoms on Earth's surface. The first part is easy: 1) Calculate surface ...
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### heron's formula proof

I have seen an interesting proof of Heron's formula here. It is very simple, but I do not understand one point. The author demands, that the formula should contain factor $(a+b+c)$, because when we ...
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### Determining area of graph based on unknown boundaries bounded by special trigonometry graphs

Interesting question for all to solve that i chanced upon luckily. let the area bounded by two graphs below from $m \pi$ to $(n-1) \pi$ be A, where $m$ and $n$ are integers. Express $n-m$ in terms ...
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### 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. ...
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### What is the area of a triangle with sides $\sqrt{5}$, $\sqrt{10}$, $\sqrt{13}$?

I found a "fun algebra problem" that asks you to find the area of a triangle whose sides are $\sqrt{5}$, $\sqrt{10}$, $\sqrt{13}$. After some algebra hell trying to work with Heron's formula, I ...
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### A given perimeter length that is circular encloses the maximum area - which are the (analytic) proofs? [duplicate]

I'm guessing Newton, because of his integrals. But what proofs have been established, and which is the most mathematically intuitive one? I was looking for the tag "circumference", supplied the newer ...
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### Finding the area between two curves.

Context: High School question. Find the surface area between the curve of the function $y=6-3x^{2}$ and the function $y=3x$ in the interval $[0,2]$ My approach: -We must find the points of ...
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### Proving an integration identity using the co-area formula

I want to prove that identity: $$\int _{B^n(0,R)} f(x)dx = \int _0^R r^{n-1} \int _{S^{n-1}} f(ry)dS(y)dr$$ where $B^n(0,R)$ is $n$-dimensional ball and $S^{n-1}$ is the $(n-1)$-dimensional sphere (...
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### Maximizing the size of n similar equilateral triangles from a rectangle

I have 3 rectangles of greenhouse sheeting material. They are each 12 feet long and 4 feet high. I want to use this material to clad a 3/4 icosahedron dome structure. What that means is that I will ...
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### Solve $\int^2_{-1} (1-x)dx$ by thinking in terms of area?

On our last quiz in Calculus 1, my professor asked us to solve $\int^2_{-1} (1-x)dx$ by thinking in terms of area. I don't know what he means by, "thinking in terms of area". I can solve it myself, ...
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### Given a set of points, find maximum area of triangle

Given a finite set of 2-d points, I need to find the maximum area of triangle formed. My solution steps : Take mean of points , lets call it (x_m, y_m). Take 3 most distant points from (x_m, y_m). ...
### Area below the curve $y=\left[\sqrt{2+2\cos2x}\right]$
Find the area below the curve $y=[\sqrt{2+2\cos2x}]$ and above the $x$-axis , $x\in [-3\pi,6\pi]$, (where $[.]$ denotes the greatest integer function) . My approach: $$y=[\sqrt{2+2\cos2x}]$$ =[\...