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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 ...
Den Ji's user avatar
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2 votes
3 answers
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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 ...
Quadrics's user avatar
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2 votes
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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 ...
Haoran Chen's user avatar
2 votes
0 answers
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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 ...
youthdoo's user avatar
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1 vote
1 answer
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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?...
kirismasdada's user avatar
0 votes
1 answer
48 views

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 ...
Xirdal's user avatar
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1 vote
1 answer
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Is there a straightforward way to triangulate this tetrahedrally-symmetric convex surface according to these criteria?

I have a tetrahedrally-symmetric surface of constant width defined in spherical coordinates by the support function $$ h(θ, φ) = \frac{S}{16} ⋅ \left(\sin(θ)^3 ⋅ \cos(3 ⋅ φ) + \frac{5 ⋅ \cos(θ)^3 - 3 ⋅...
Lawton's user avatar
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0 votes
1 answer
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Inscribing a square bar in a box

Question: You're given a box (a cuboid) of known dimensions $2 a \times 2 b \times 2 c $. And you have a bar with a cross section that is a square of known dimension $d$. You want to inscribe the ...
Quadrics's user avatar
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1 vote
1 answer
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Find the algebraic equation of the ellipsoid of known orientation inscribed in a cuboid of known dimensions

Given a cuboid of dimensions $2 a \times 2 b \times 2 c$, and given a $3 \times 3$ rotation matrix $R$, I want to inscribe an ellipsoid whose axes are respectively along the directions specified by ...
Quadrics's user avatar
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1 vote
1 answer
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Finding the parametric equation of a $3D$ ellipse that is inscribed in a cuboid

Suppose you're given a cuboid centered at the origin, and with its edges parallel to the coordinate axes. Suppose its measure is known to be $2 a \times 2 b \times 2 c $ along the $x, y, z$ ...
Quadrics's user avatar
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What is the substantial generalization of the “acuteness” to 3-simplex?

A 2-dimensional simplex is just a triangle, and a 3-dimensional simplex is just a tetrahedron… therefore for convenience, I simply use the term triangle and tetrahedron in the following words. We know ...
user688486's user avatar
1 vote
1 answer
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Real usage of “pure” quaternions in stereometry?

There are two major categories of the "quaternions". It is well-known that a (nonzero) versor represents a three-dimensional rotation operator. A versor is a unit quaternion or a normalized ...
user688486's user avatar
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0 answers
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3D analogy of 2D size maximization problem

Pardon me, but here is another problem which surfaces from my own extension on a solved problem, i.e. to say there is no "answer key" to refer to, for the problem I am proposing. I tend to ...
Rupen Kohli's user avatar
1 vote
0 answers
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Is there any rigorous proof of the generalized Pappus's area theorem in ℝ³?

The little-known Pappus's area theorem in Euclidean plane geometry can “be thought of as a generalization of the Pythagorean theorem”. However, the Problem 34 in Nicholas Donat Kazarinoff's Geometric ...
user688486's user avatar
8 votes
2 answers
177 views

Probability of each type of inscribed octahedron

Fix a $V\in\mathbb{N}$ with $V\ge 4$. Randomly pick $V$ points on a sphere (independently and uniformly with respect to the surface area measure). You may think of the convex hull of these $V$ points. ...
Jeppe Stig Nielsen's user avatar
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2 answers
41 views

perpendicular lines through $F$ project to perpendicular lines

Let $C(0,0,\sqrt{2})$ and $F(0,\sqrt{2},0)$ be two points in $\Bbb R^3$. $AF,BF$ are arbitrary perpendicular lines through $F$ on the plane $y=\sqrt{2}$. These lines project to the lines $DF,EF$ under ...
hbghlyj's user avatar
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2 votes
1 answer
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If seven vertices of a hexahedron lie on a sphere, then so does the eighth vertex.

I'm trying to prove https://imomath.com/index.cgi?page=inversion (Problem 11) by projective geometry: If seven vertices of a (quadrilaterally-faced) hexahedron lie on a sphere, then so does the ...
auntyellow's user avatar
2 votes
1 answer
73 views

A set of lines, each of which intersect the others, are either coplanar or share a single common point.

Prove that a set of lines, each of which intersect the others, are either coplanar or share a single common point. Source: Hadamard This is a surprisingly difficult problem, because it requires ping-...
SRobertJames's user avatar
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0 votes
3 answers
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How to find missing vertex/point of a tetrahedron

I need to find the missing point of a tetrahedron side length $x$, with three points being $$v_1=(0,0,0) \quad v_2=\left(\frac{1}{2}x,0,\frac{\sqrt3}{2}x\right) \quad v_3=(x,0,0)$$ I can't seem to ...
SethRayCarlsen's user avatar
0 votes
1 answer
47 views

Finding the equation of two lines lying on the surface of a hyperbloid

Trying to solve this question: Surface S is obtained by revolving the line $x^2-z^2=4$ around the "z" axis. Write the equation for S. Show that exactly two lines pass through M=(2,0,0) ...
TEGNO's user avatar
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2 votes
1 answer
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Can space be augmented with a plane at infinity so that parallel planes intersect at a line at infinity?

The real plane can be augmented with a line at infinity such that two parallel lines intersect at a point at infinity, and the set of all such points forms a line. In space (3 dimensional solid ...
SRobertJames's user avatar
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2 votes
1 answer
53 views

Can a oblique antiprism be constructed?

Can a oblique antiprism be constructed? Intuitively, it would seem oblique antiprism exist: Take any right antiprism and translate one of the parallel faces within it's plane. I'm baffled, though, ...
SRobertJames's user avatar
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12 votes
3 answers
3k views

What's wrong with this derivation of the volume of a hemisphere?

My idea to calculate the volume of the hemisphere is to sum up the area of circles of all radii up to the radius of the hemisphere we are interested in: $$\int_0^r \pi x^2 dx$$ This gives $\frac{1}{3}\...
timtam's user avatar
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2 votes
0 answers
56 views

Can the construction of this 2D Curve of Constant Width be adapted to a 3D Surface of Constant Width?

A Surface of Constant Width is a 3D surface with the special property that any two parallel planes which are tangent to it are always a constant distance apart, no matter the relative rotations of the ...
Lawton's user avatar
  • 1,861
0 votes
1 answer
172 views

Geometrically determining the symmetry of a tetrahedron (using solid geometry, not group theory)

It is well established that a regular tetrahedron has 12 orientation preserving symmetries, the group $A_4$. To better understand these symmetries, I set out to identify them geometrically: Prove that ...
SRobertJames's user avatar
  • 4,450
2 votes
0 answers
57 views

In a tetrahedron, reflect each point over the line from an edge's midpoint through its center. Is it the tetrahedron itself?

Let $T$ be a regular tetrahedron, and $\ell$ a line from the midpoint of one edge $E_1$, through the center $C$, to the midpoint of another edge $E_2$. How do I prove that the reflection of each point ...
SRobertJames's user avatar
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0 votes
0 answers
51 views

The intersection of $ n $ cylinders in $ 3$-dimensional space

I recently found out about the Steinmetz Solids, obtained as the intersection of two or three cylinders of equal radius at right angles. If we set the radius $ = 1 $ the volumes are respectively $ V_2 ...
user967210's user avatar
1 vote
1 answer
49 views

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 ...
Jacco's user avatar
  • 125
2 votes
1 answer
137 views

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 ...
SRobertJames's user avatar
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0 votes
0 answers
47 views

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. ...
Cyril's user avatar
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1 vote
1 answer
126 views

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 ...
trying's user avatar
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0 votes
0 answers
24 views

How does the angle between tangent vectors of a curve at different points and a given fixed uniform vector field change on rotation of curve?

I have to evaluate an integral $\int B\ dl \sin\theta$ for quarter $CD$ of the given ring of radius $r$. [$\theta$ being angle between tangent vector($d\vec l$) and $\vec B$] In the case where $\...
Aurelius's user avatar
  • 471
2 votes
1 answer
71 views

Prove that all three bisecting planes of the dihedral angles of a trihedral angle intersect along one straight line.

Prove that all three bisecting planes of the dihedral angles of a trihedral angle intersect along one straight line. I attempted like this we can take one triangle at first as all its bisectors will ...
Tutor4872's user avatar
1 vote
2 answers
51 views

finding the height of a prismoid

Imagine a prismoid container like the one pictured. I have poured water into it and I know all values of the resulting water prismoid - I know its volume, it's height, the areas of its bases, the area ...
morkfromork's user avatar
2 votes
0 answers
70 views

Surface (superior and lateral) and volume of an ungula

Context Definition: An ungula is the solid obtained by cutting a cone with a plane and keeping the part between the base of the cone and the plane I couldn't find the formulas to obtain the upper ...
Math Attack's user avatar
1 vote
1 answer
80 views

Solving a railroad fill problem based on a prismatoid.

I need help what am I doing wrong The railroad fill shown in the figure has sloping sides which rise vertically 0.5 feet for each foot horizontally. The top of the fill ABCD is horizontal and the ends ...
Den Ji's user avatar
  • 99
2 votes
0 answers
48 views

Calculating the volume of a corner cut from the base of a right circular cone

I'm trying to find the volume of some arbitrary concrete tetrapod (the things they use to break waves on a beach) and I've broken the problem into a few parts. There's the 4 partial cones and the ...
Golem's user avatar
  • 21
5 votes
3 answers
122 views

Winding a wire of definite width around a right cylinder

The original question is presented like this: A copper wire, $3~mm$ in diameter is wound about a cylinder whose length is $12~cm$ and diameter $10~cm$, so as to cover the curved surface area of the ...
Darshit Sharma's user avatar
2 votes
1 answer
158 views

Calculating material needed for a lamp shade with an allowance for the seam. [closed]

I wanted to solve this problem but my English is bad and not familar with tailor stuff.... A lamp shade is in the form of a frustum of a cone with slant height 7 inches, radii of bases 3 inches and 7 ...
Den Ji's user avatar
  • 99
0 votes
0 answers
22 views

Do these formula applies to Frustrum of a right pyramid which have a regular polygon as a base?

So here is the problem: A baking pan has a rectangular base 12 inches by 8 inches; the sides and ends of the pan slope outward, so that the upper edges measure respectively 13.5 inches by 9 inches. If ...
Den Ji's user avatar
  • 99
0 votes
0 answers
28 views

Slight Confusion about Spherical Sector Definition

A book that I am reading defines the spherical sector as follows: A spherical sector is a solid generated by rotating a sector of a circle about an axis which passes through the center of the circle ...
Niko's user avatar
  • 73
0 votes
1 answer
27 views

Relationship of a section cut parallel to a base of a pyramid and its height proof

Is there an algebraic proof for this? I was trying to solve how this relationship was made. Thanks! Also, what is the proper name for this relationship?
Den Ji's user avatar
  • 99
2 votes
0 answers
99 views

Seeking guidance on calculating the volume of a four-dimensional cone

I'm working on a problem involving a four-dimensional cone defined as follows: $C = \left\{\left(x,y,z,t\right)| (x,y,z) \in (1 - \frac{t}{12})B,0 \leq t \leq 12\right\}$ where the base $B$ is ...
Ivn's user avatar
  • 119
0 votes
1 answer
43 views

Find the perimeter of the section determined in the cube, by a plane guided by 3 points determined below

A cube $ABCD A_1 B_1 C_1 D_1$ is given, with edge $a$. On the edge $C_!D_1$we take a point $L$, with $C_1L=\dfrac{3a}{4}$; on the edge $A_1B_1$ we take point $M$, with $A_1M=\dfrac{a}{2}$; on the edge ...
peta arantes's user avatar
  • 6,991
1 vote
1 answer
45 views

Equation of inscribed ellipsoid in a parallelepiped given $2$ tangency points

Given three vectors $u_1, u_2, u_3 \in \mathbb{R}^3$ that are linearly independent, you build a parallelepiped by specifying a vertex $V_1$, and then the other $7$ vertices follow: $V_2 = V_1 + u_1 $ $...
Quadrics's user avatar
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2 votes
0 answers
57 views

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 ...
Next-Door Tech's user avatar
1 vote
0 answers
100 views

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,...
VieteaJumper's user avatar
4 votes
0 answers
103 views

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 ...
Ed Pegg's user avatar
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1 vote
2 answers
258 views

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 $T$...
IONELA BUCIU's user avatar
1 vote
1 answer
73 views

Calculation of the volume of Steinmetz solid

According to https://mathworld.wolfram.com/SteinmetzSolid.html, the volume of the Steinmetz solid with radii 1 referencing (11) is $V_2(1,1)=\displaystyle\int_{-1}^{1}\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}...
ThomasL's user avatar
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