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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Why is the derivative of a circle's area its perimeter (and similarly for spheres)?

When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle. Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is ...
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Find the area of largest rectangle that can be inscribed in an ellipse

The actual problem reads: Find the area of the largest rectangle that can be inscribed in the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.$$ I got as far as coming up with the equation for ...
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Circle areas on squared grid

There is a circle. On 9 equal squares. Every square has some value assigned to it. Every square gets weight, depending of what percentage of it is circle (area-wise). I need to find circle radius, ...
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How calculate the shaded area in this picture?

Let the centers of four circles with the radius $R=a$ be on 4 vertexs a square with edge size $a$. How calculate the shaded area in this picture?
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Which area is larger, the blue area, or the white area?

In the square below, two semicircles are overlapping in a symmetrical pattern. Which is greater: the area shaded blue or the area shaded white? My Solution Let the length of each side of the square ...
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Find the area where dog can roam [duplicate]

A dog is tied to circular pillar by a rope. Radius of this pillar is $1m$ and length of rope is $\pi m$. What is an area where dog can roam? I tried to find the area of all semicircles and then to ...
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Do two right triangles with the same length hypotenuse have the same area?

I watched computer monitors and I asked myself, do two monitors with the same display diagonal have the same display area? I managed to find out that the answer is yes, if two right triangles with ...
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Area bounded by $\cos x+\cos y=1$

What is the area of the region $\cos x+\cos y > 1$, where $|x|,|y|<\pi$? In other words, is there a "closed" form -- using functions that are well-known and nice to work with -- for this ...
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Reasoning behind the cross products used to find area

Alright, so I do not have any issues with calculating the area between two vectors. That part is easy. Everywhere that I looked seemed to explain how to calculate the area, but not why the cross ...
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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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Finding number of “Pixels” in a Circle using diameter

I'm trying to figure out how to calculate the number of whole pixels in a pixel circle using the diameter of the circle. I understand how to find the area of a circle using diameter. But I'm ...
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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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What is the chance that a PDF with compact support is concave?

Relevant questions and answers, in chronological order: When do equations represent the same curve? Find a smooth function with prescribed moments Does a sequence of moments determine the function? ...
I have some troubles with this problem : Let $ABCD$ be a convex quadrilateral. $M$, $N$, $P$ and $Q$ are the midpoints of the sides $AB$, $BC$, $CD$ and $AD$. $AN$, $BP$, $MD$ and $CQ$ are ...
area-preserving iff $|\det |=+1$
Why is a (not necessarily linear) mapping $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ area- and orientation preserving iff the determinant of its jacobian is $\pm 1$ ? (I understand by an area-...