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141 votes
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What is the maximum volume that can be contained by a sheet of paper?

This problem reminds me of tension field theory and related problems in studying the shape of inflated inextensible membranes (like helium balloons). What follows is far from a solution, but some ...
user7530's user avatar
  • 49.5k
115 votes

Calculating the volume of a restaurant take-away box that is circular on the bottom and square on the top

A different approximation could be to take the cross-section at each height not as a linear interpolation between top and bottom surface, but as squares with rounded corners. This fits the photograph ...
AlienAtSystem's user avatar
73 votes
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Calculating the volume of a restaurant take-away box that is circular on the bottom and square on the top

As the shape of the solid is not clearly defined, I'll make the simplest assumption: lateral surface is made of lines, connecting every point $P$ of square base to the point $P'$ of circular base with ...
Intelligenti pauca's user avatar
61 votes
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How deep is the liquid in a half-full hemisphere?

Assuming the spoon is a hemisphere with radius $R$, let $x$ be the height from the bottom of the spoon, and let $h$ range from $0$ to $x$. The radius $r$ of the circle at height $h$ satisfies $r^...
J. W. Tanner's user avatar
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58 votes
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Matrix determinant contradicts corresponding box volume – how is it possible?

The given vectors do not form a cube, so using the cube's volume formula is invalid. In particular, the vectors would need to be orthogonal (dot product zero for all pairs) for said formula to be ...
Parcly Taxel's user avatar
55 votes

Paradox about the volume of a cylinder

Where is the error in my previous argument based on infinitesimals? The error is here: Therefore, the volume of the cylinder can be thought as made out of an infinity of areas of such rectangles ...
Martin Rosenau's user avatar
51 votes
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Why isn't the derivative of the volume of the cone its surface area?

If you continuously enlarge a solid cone by adding material to the base, then every inch of added height corresponds to an inch-thick layer added to the circular base, so the rate of change of volume ...
Karl's user avatar
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46 votes
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How to find a volume of this figure (which is $3080 \text{ cm}^3$) in a few seconds?

I looked at the horizontal layers. Top layer has seven, and each layer below shows seven more. So the number of cubes is $$ 7+14+\cdots+70=\frac{77}2\cdot10=385\,. $$
Lubin's user avatar
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41 votes
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Is it simply a coincidence that if you differentiate the formula for the volume of sphere you get the formula for the surface area of sphere?

Start with a sphere of radius $r$. Now let the radius of the sphere grow by some tiny amount $\Delta r$. How much has the volume changed? By the definition of the derivative, it has changed by ...
Arthur's user avatar
  • 201k
40 votes

What is the maximum volume that can be contained by a sheet of paper?

This is certainly not optimal, but comparatively straightforward to calculate and a moderate improvement over the previous solutions. If we let the sides of the box fall outward, the top rectangular ...
joriki's user avatar
  • 239k
36 votes

Matrix determinant contradicts corresponding box volume – how is it possible?

Your 3 vectors are related to a cube, but they do not form the cube edges. They are the diagonals of 3 adjacent faces:
Eric Duminil's user avatar
  • 3,494
36 votes

How deep is the liquid in a half-full hemisphere?

It may be surprising that the problem actually admits an analytic solution. A spherical cap is the difference between two overlapping cones, one with a spherical bottom and the other with a flat ...
Quanto's user avatar
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32 votes
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Paradox about the volume of a cylinder

The rotation contributes to the volume by $2\pi R$ only for the side of the rectangle that is opposite to its rotation axis. The circumference covered by each point of the rectangle depends on its ...
Taroccoesbrocco's user avatar
31 votes
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Intuition for volume of a simplex being $\frac 1{n!}$

It's easier to see an alternate simplex has volume $1/n!$: The set of all points $(x_1,x_2,\dots,x_n)$ with $0\leq x_1\leq x_2\leq\cdots\leq x_n\leq 1$. That's because the volume measures the ...
Thomas Andrews's user avatar
26 votes

Matrix determinant contradicts corresponding box volume – how is it possible?

I made a gif that shows the transformation. At the beginning you see the unit cube and at the end of the animation you see the unit cube as transformed by the matrix $$\begin{bmatrix}1 & 0 & 1\...
AccidentalTaylorExpansion's user avatar
25 votes

How deep is the liquid in a half-full hemisphere?

It makes things a bit simpler if we turn your measuring spoon upside down, and model it as the set of points $\{(x,y,z):x^2+y^2+z^2=1, z\ge 0\}$. The area of a cross-section at height $z$ is then $\pi(...
TonyK's user avatar
  • 65k
21 votes
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What is the volume of the $3$-dimensional elliptope?

The integral can be separated: $$I = 2\int_{-1}^1 \sqrt{1-x^2} dx \cdot \int_{-1}^1 \sqrt{1-y^2} dy = 2\left(\int_{-1}^1 \sqrt{1-t^2} dt\right)^2$$ This integral is straight-forward using the ...
David P's user avatar
  • 12.3k
21 votes
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Do balls of finite radius have finite volume?

Consider the open unit ball in $\mathbb{R}^2$ with polar coordinates $(r,\theta)$ and a metric of the form $g=dr^2+f^2(r)d\theta^2$, where $f$ is a smooth positive function chosen so that $g$ is well ...
Kajelad's user avatar
  • 15k
20 votes
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Looking for where I went wrong: Finding the volume of a solid that lies within both a cylinder and sphere

“Isn’t $r^2$ always 1?” Yes, on the surface of the cylinder. But you aren’t integrating on the surface of the cylinder. You’re integrating over a three-dimensional region, namely the one bounded by ...
symplectomorphic's user avatar
18 votes

Why is the volume of a cone not half the volume of the cylinder of which it is a part?

The parts of the triangles on the outside contribute more to the volume than the parts on the inside, and the triangles belonging to the cone are on the inside, so they contribute less than half the ...
Vercassivelaunos's user avatar
16 votes
Accepted

Calculate Volume of Torus Given Circumferences

By Pappus' centroid theorem, the volume of a torus generated by the rotation of a circle with radius $r$ with its centre on a circle with radius $R$ is just given by $2\pi^2 r^2 R$. In our case we ...
Jack D'Aurizio's user avatar
16 votes
Accepted

How to check if a sphere passes through another sphere when both travels in a straight line through 3d space

First of all, your problem is actually 1-dimensional, because the $y$ and $z$-coordinates of either ball will be zero at any time. As the balls are approaching each other, they will always intersect! ...
emacs drives me nuts's user avatar
15 votes

Intuition for volume of a simplex being $\frac 1{n!}$

Denote the volume of this simplex by $\sigma_n$. Foliating the simplex with "horizontal" hyperplanes $x_n=z$ $(0\leq z\leq 1)$ and applying Fubini's theorem we obtain $$\sigma_n=\int_0^1(1-z)^{n-1}\...
Christian Blatter's user avatar
15 votes

What is the maximum volume that can be contained by a sheet of paper?

This may not be the optimum. But an easy solution with larger volume than the box. 1.14228 l. (Of course we need to tape it at base to hold)
AppoopanThaadi's user avatar
15 votes

Calculating the volume of a restaurant take-away box that is circular on the bottom and square on the top

Since it differs in circumference Are you sure about this? One thing to note about the box is are the vertical stripes. Your image suggests that they have equal width and I see production- or design-...
Wrzlprmft's user avatar
  • 5,748
14 votes

How does a 0-ball have volume 1?

$0$-dimensional space is just a single point and every ball of positive radius contains that point. Moreover, the measure in this space is just the counting measure. So the volume of the ball is $1$ ...
Hagen von Eitzen's user avatar
13 votes

Scaling of Lebesgue measure under a linear transformation and the volume of a parallelepiped.

$\def\vect{\mathbf} \def\diag{{\rm{diag}}} \def\R{\mathbb R} \def\vol{{\rm vol}} \def\sign{{\rm sign}} $ There are two ideas involved: One is that uniqueness of Lebesgue measure as a translation ...
fred goodman's user avatar
  • 4,283
12 votes
Accepted

Is orientability needed to define volumes on riemannian manifolds?

You don't need orientability to define the volume of a parametrized region. Your value $vol(R)$ is completely well-defined in terms of a parametrization, and doesn't even depend on the parametrization ...
Aloizio Macedo's user avatar
  • 34.4k

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