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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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 ...
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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, ...
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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 ...
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Geometric analysis of the motions of a tetrahedron

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 ...
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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 ...
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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 ...
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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. ...
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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 ...
trying's user avatar
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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 $\...
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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 ...
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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 ...
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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
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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 ...
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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 ...
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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
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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 $ $...
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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 ...
Next-Door Tech's user avatar
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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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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}...
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Generalising Thales theorem for points on a sphere to form a 3-orthoscheme (tetrahedron.)

I am trying to find the condition that four points $p_1,p_2,p_3,p_4$ on the unit sphere $\mathbb{S}^1$ need to statisy in order to form a 3-orthoscheme (Tetrahedron with all faces as right angled ...
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Can you tell me about the (probably) well known relationship between the coefficients of a cubic and some features of a rectangular solid?

If we look at the expansion of this $$(x+a)(x+b)(x+c)=x^3+(a+b+c)x^2+(ab+bc+ca)x+abc$$ And consider a rectangular solid with length, width and height of $a, b, c$ respectively. Then $$l_{edges}=4(a+b+...
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Similar Reflection Tetrahedron

With a few intersection points of heptagon diagonals, a heptagon with double the area can be constructed. The reflection triangle of the heptagonal triangle (in green) is another heptagonal triangle (...
Ed Pegg's user avatar
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Tangential tetrahedron

For a tetrahedron, the circumsphere is calculated, then it's recentered at the origin and the tangent planes at the vertices are found. Plane intersections are found, then the recentering is reversed. ...
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Are there four dimensional generalizations of the Reuleaux triangle and other solids of constant width?

Is there a four dimensional generalization of the Reuleaux triangle? What is it called, and what properties does it have? Thank you!
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Find the volume and total surface area of the solid formed

Let the planes be defined by $$|x|+|y|+|z|=1$$ Find the volume of the solid enclosed and the total surface area of the solid thus generated. I am not able visualise the solid. What will it be$?$ I ...
MathStackexchangeIsNotSoBad's user avatar
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Projections of zonohedra

I noticed that any generic orthogonal projection of a cube has exactly two crossings. This led me to wonder about generalizations. A friend suggested I look at zonohedra. A zonohedron is the ...
Akiva Weinberger's user avatar
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Geometric properties of Snub Icosidodecadodecahedron and Medial Hexagonal Hexecontahedron ($U46$ and $U46'$)

I would like to calculate the closed form of some values ​​relating to $U46$ and $U46'$ (especially angles and volumes). I found this site where the values are given in term of root of equations ...
Math Attack's user avatar
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Closed form of volume of Small Retrosnub Icosicosidodecahedron $(U72)$ and Great Dirhombicosidodecahedron $(U75)$

I would like to calculate the closed form of the volume of the following solids: $U72$ (known as Small Retrosnub Icosicosidodecahedron) $\approx 0.2286299526 l^3$ $U75$ (known as Great ...
Math Attack's user avatar
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Volume of half of a truncated cylinder

I'm having trouble working out what a shape is called, let alone how to calculate its volume. I'm looking for a way to calculate the volume of a shape derived from a truncated cylinder, as described ...
R Mooney's user avatar
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How can I estimate the volume of a solid object, knowing only it's longitudinal corss-sectional area?

Let's say the shape is too complex to split it into simpler parts and solve it analytically. I can obtain it's longitudinal cross-sectional area by loading the image into an image editor, scaling it ...
John Smith's user avatar
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1 answer
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Determine the ray which passes through the center of a sphere given its projection on the image plane.

I am not a mathematician, but I know that a sphere projected on the image plane becomes an ellipse under perspective transformation. I don't know whether the ray starting from the center of projection ...
NMO's user avatar
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parametric equation for curve of intersection of two cylinders

I'm struggling to find a way to express the curve of intersection of two surfaces. Context: this is for a university assignment about the MuPAD CAS system. It's a correspondence university and I'm not ...
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Dehn Invariants of two space-filling tetrahedra

Here are vertices for two space-filling tetrahedra. $A = ((0, 0, 2),(0, 4, 2),(1, 2, 2),(2, 2, 0))$ $B = ((0, 3, 3),(1, 4, 2),(3, 2, 4),(1, 0, 2))$ Dihedral angles for A: {Pi/4, ArcCos[-(1/Sqrt[6])], ...
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What is the probability that a convex polyhedron will land on each of its faces if thrown as a die?

Given an arbitrary convex polyhedron that we are using as a die, is it possible to know the probability of it landing on each face? I am assuming that these probabilities depend on the shape of each ...
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Angle of tilt of a rectangular water tank

A cuboid tank is placed in the $xy$ plane, with its base centered at the origin. The base rectangle measures $5$ along the $x$ axis, and $7$ along the $y$ axis. The height is $9$. It is filled to $\...
of course's user avatar
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Largest circle inscribed in a cube

Given a cube of side length $a$, what is the the radius of the largest circle that can be inscribed in the cube? My Attempt: I just assumed that the largest circle is the incircle of the hexagon ...
of course's user avatar
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13 votes
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Maximum volume enclosed by a piece of cloth in the shape of a unit circle

I have a piece of cloth in the shape of a unit circle (or disk), a needle and thread. I want to stitch the cloth together so that it can contain stuff (say, lots of small beans) and the stuff cannot ...
Dan's user avatar
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Find the constraints which guarantee that 6 planes in 3D space form a convex box containing the origin?

I ever asked a question to find the constraints which ensure 4 lines in 2D space form a convex quadrilateral; see link and it has been solved by @YNK author perfectly. Now I hope to extend it and came ...
Pat_Guangtailang's user avatar
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Solid Geometry Problems [closed]

Please help me. I'm curious how to solve it. Given the cube $ABCD.EFGH$. If $P$ is the point of extension of $HG$, so that $HP:HG=3:2$, then the angle tangent between the lines $AP$ and $HB$ is $\...
Dom Kang's user avatar
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1 answer
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Find the relation in which the plane divides the edge of the tetrahedron

The tetrahedron ABCD is given . The points M, N and K lie on the edges AD , AB and BC, respectively, and AM:MD = 2:3 , BN:AN = 1:2 and BK:KC = 1:1. Construct a section of the tetrahedron with the MNK ...
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