# Questions tagged [solid-geometry]

In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space. (Ref: http://en.m.wikipedia.org/wiki/Solid_geometry)

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### Why there is two different answers for the volume of a frustum?

Here is the problem: A margarine tub has the shape of the frustum of a cone. With the lower base having diameter length 11 cm and the upper base having diameter length 14 cm, the volume of such a ...
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### Determine the inscribed ellipsoid within a cube with given ratios of axes

Given a cube centered at the origin, with side length $2a$, determine the length of the semi-axes of the ellipsoid inscribed in the cube, touching all its $6$ faces, such that the semi-axes lengths ...
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### Relationship between angles at Fermat point in high dimensions

It is well known that at the Fermat point of a triangle (no angle $\geq 120$ degrees), the three angles are all $120$ degrees, or $\frac{2\pi}{3}$. In 1 dimension, the Fermat point of a segment is not ...
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### Maximize volume of tetrahedron $ABCD$ given $AC=AD=BC=BD=1$

In tetrahedron $ABCD$, the edges $AC$, $AD$, $BC$, $BD$ are all of length $1$. Find the maximum value of the volume of the tetrahedron. Here is my solution: let $CD=x$, then $x\in(0,2)$ by the ...
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1 vote
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### Height of a pyramid formed from a random triangle

I have drawn a height (brown) from the orthocenter of the triangle, this height is perpendicular to both side edges, is it possible to relate this height to a,b and c? if possible, what is the formula?...
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### Origin of the round square triangle puzzle

I need to refer, in a research paper, to the solid mentioned in this old thread: Is there a name for a 3D shape that looks like a circle when viewed from one axis, a square from another, and a ...
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### Finding the 5-trapezohedron height to width ratio for perfect midsphere

WolframAlpha shows an example image for a midsphere in a 5-trapezohedron. The example image shows an 5-trapezohedron where the midsphere perfectly touches all 4 edges of each face. However, the shape ...
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### When can a single plane $P$ be constructed through six lines $\ell_n$ in different planes?

In $\mathbb R^3$, consider the reference planes $x = 0, x = 1, y = 0, y = 1, z = 0, z = 1$, and one line in each plane: $\ell_{x=0}, \ell_{x=1}, \ell_{y=0}, \ell_{y=1}, \ell_{z=0}, \ell_{z=1}$. For ...
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### Finding a tetrahedron with known sides of the base and the angles opposite to these sides of the side faces.

Given a tetrahedron ABCD with known sides $a, b, c - AB, AC, BC$ of the base and the angles $\alpha, \beta, \gamma - \angle{ADB}, \angle{ADC}, \angle{BDC}$ of the side faces opposite to these sides. ...
1 vote
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### Probability of flipping a Hershey's Kiss and it landing on its base

I, like many other AP Statistics students, just spent an entire class flipping Hershey's Kisses and trying to determine the empirical probability of landing it on its base. This led me to wonder if ...
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### What is a solid hemisphere formally called?

Quick question - if the formal name for the solid counterpart of a sphere is a ball, and the formal name for a "half-sphere" is a hemisphere, what is the formal name for the solid ...
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### Analogy of Trillium Theorm

I am thinking about, is there analogy beetween 2d and solid geometry about Trillium theorm. (But im intrest in only acute or right triangles) Trillium theorem in 2d geometry holds: Let ABC be triangle,...
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### The 18 golden rational tetrahedra

In 2020, the 59 sporadic rational tetrahedra were identified. More recently, I found exact solutions for all of them. Most of them don't pair up well in terms of similar triangles that would allow ...
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### Prove some results in a cube...

Question Let $ABCDA'B'C'D'$ be a cube and the points $M, N, Q$ the means of the sides $A'B', A'D', DC$. We denote by $\alpha=(MNQ)$. a) If the line $D'C'$ intersects the plane $\alpha$ at the point $T$...
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