# Questions tagged [geometry]

For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, and angles.

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### Number of unit cubes meeting the boundary of a convex set

Suppose $C \subseteq [0,n)^k$ is a convex set, and $\partial C$ is its topological boundary: its closure minus its interior. Is it true that $\partial C$ meets at most $2k n^{k-1}$ unit cubes? By a ...
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### Find a partial area of a trapezoid

The formula for the area of a trapezoid is $$A = \frac{(a+b)}{2}h$$ where a and b are the length of each base and h is the trapezoid's height. So I want to figure out the area of a portion of the ...
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### Estimates on Covering Number for Convex Polytopes Partitioning a Convex Set

Consider a convex set $K\in\mathbb{R}^3$ and a collection of convex 3-polytopes $C^i\subseteq K$ of equal volume $\frac{1}{N}$ for $i\in(1,\ldots,N)$ that partition $K$ (a Voronoi or Laguerre diagram ...
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### The "seashell constant": closed form for $\frac12\exp\int_0^1-\log(\sin(\frac{\pi}{6}+\frac{2\pi}{3}x))\mathrm dx$?

I am looking for a closed form for $R=\frac12\exp\int_0^1-\log\left(\sin\left(\frac{\pi}{6}+\frac{2\pi}{3}x\right)\right)\mathrm dx\approx0.6159$. Wolfram does not give a closed form for $R$. Wolfram ...
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### The diagonals of a regular pentagon with a side length of 1 form a new, smaller regular pentagon. What is the side length of the smaller pentagon?

The diagonals of a regular pentagon with a side length of 1 form a new, smaller regular pentagon. What is the side length of the smaller pentagon? Attempt: I lebel eveything, but I am unable to find ...
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### Find the equation of the circle that is tangent to 2 lines

Let the lines $(d_1):4x+7y-37=0$ and $(d_2):7x-4y+49=0$, and the points $A(4;3)$ and $N(18;-5)$. Let $\mathcal C$ the circle that is tangent to $(d_1)$ and $(d_2)$, the intersection between $(d_1)$ ...
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### Find the minimum and maximum values of $f(P ) = \angle ABP + \angle BCP + \angle CDP + \angle DAP$

Let $P$ be a point inside or on the boundary of a square $ABCD$. Find the minimum and maximum values of $f(P ) = \angle ABP + \angle BCP + \angle CDP + \angle DAP$. calling the angles $a,b,c,d$ if ...
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### prove the addition identity cos(x+y) using the law of cosines [closed]

I’ve drawn a triangle with sides a b and c and written down two equations for side c using the law of cosines with angle x+y and using the sine of angle x and sine of angle y. I don’t know how to ...
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### Shortest path on the surface of a cylinder between given points $A$ and $B$

Suppose you have the cylinder $x^2 + y^2 = R^2$ And points $A = (R, 0, 0)$ and $B = (0, R, h)$. Find the parametric equation of the curve of shortest length connecting $A$ and $B$. My attempt: If ...
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### Determine the rotation necessary to bring a plane in contact with an ellipsoid

Given the ellipsoid $(r - C)^T Q (r - C) = 1 \tag{1}$ And the plane $n^T (r - r_0) = 0$. I want to determine the angle of rotation about an axis whose unit direction vector is $a$ and passes ...
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### Derivation of the volume of a half-sphere

I am trying to derive the volume of a half-sphere with a constant r using integration. Integration I first try to integrate 90 degrees from the top of the halfsphere down to the bottom plane of the ...
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### Geometry question on area lemma or ratio and proportion theorem

Let ABC be a triangle and D, E are points on the segment BC, CA respectively, such that AE = λAC and BD = μBC. Let AD, BE intersects at F. Find, in terms of λ and μ, the ratio AF : FD.
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### Is it possible to find r in terms of the other variables?

I've never posted before and I'm sorry if this kind of question isn't suited to this forum. In the image I've attached, I'm wondering if it's possible to find the length r in terms of the lengths a ...
1 vote
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### How to find the shortest line segment connecting two skew lines, with constraint that line is parallel to some plane?

I've found lots of examples for how to identify the shortest line segment connecting two skew lines. For example, here: Distance between Skew lines However, for purposes of a simple game engine I am ...
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### Intersection of planes in a 3d space WITHOUT vectors [closed]

We are learning about finding the intersection of planes / lines in 3D space by drawing diagrams, and I don't get it at all. The problem is we aren't doing anything with vectors or graphical diagrams, ...
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### Center of gravity of slanted cylinder

Trying to solve the problem of the buckling of a water storage tower. As the storage tank displaced, the center of gravity/mass moves. Water which once occupied a cylindrical volume is now occupying ...
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### Strange substitution made in a paper to find asymptotics

In the quoted section from this paper, why is the author able to "substitute this result into Eq. (2.1)"? This should hold for $z$ large. But not everything on the contour is large. Why can ...
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### How to calculate distance of diameter on rounded rectangle? [closed]

I'm sorry that my reputation is not enough on this site, it is not available to upload images. I would like to know the calculation which calculates diameter of rounded rectangle. Width, length and ...
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