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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58 views

A geometry problem with a cube (solid geometry)

I ask you to solve this problem, because in the book and I have different answers Point $M$ is the midpoint of the edge $AB$ of the cube $ABCDA_1B_1C_1D_1$. Find the distance between the straight ...
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Addition formulas for direction cosines?

I will quickly explain what I am expecting from this question. Given a point $p=(x,y)$ in the plane, we can look at the ratios $$A_1(p)=\dfrac{x}{r},\qquad A_2(p)=\dfrac{y}{r},$$ where $r$ is the ...
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5answers
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Intuitively understanding why the volume of a cylinder of height $h$ and radius $R$ is $\pi R^2h$

It's well known that a cylinder with height $h$ and the base radius $R$ will have a volume of $$V = \pi R^2h$$ I'm trying to derive an intuitive understanding of it. I have two ways, one seems to be &...
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2answers
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Intersection of perpendicular chords: is this true for a sphere?

It is true that in a circle with radius $R$, if the intersection of any two perpendicular chords divides one chord into lengths $a$ and $b$ and divides the other chord into lengths $c$ and $d$, then $$...
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1answer
41 views

Three mutually perpendicular chords in a sphere with radius R [closed]

In a sphere, with radius R, there are three mutually perpendicular chords. They intersect at a point P (not the center). Point P divides the three chords into segments: a, b, c, d, e, f. Is it true ...
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1answer
20 views

volume of hyeprcube with sum of coordinates larger than x

Consider an hypercube in $N$ dimensions with one vertex in the origin and another vertex in the point with all coordinates equal to 1, with edges alignes with axes. Is there a general analytic ...
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1answer
28 views

Name of the solid built on a sphere's surface

I don't know the name of the solid built on a sphere's surface. I mean, given a sphere of radius r1, create another sphere with the same origin and radius $r_2$, with $r_2 > r_1$. I'm interested in ...
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1answer
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Finding the centre of mass of two right cones joined combined

The following question is given in my textbook Two uniform cones with base radius $r$ are joined together by their plane faces.Their lines of symmetry are aligned, the height of one cone is $6r$ and ...
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1answer
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Which solid geometrical figures can be inscribed on a sphere such that every vertex is equally distant from every other vertex inside the sphere. [closed]

I came through this question in my High School Math logic book. Ok, so one of the figure is a regular tetrahedron which can be inscribed. I was searching for other figures but didn't get one. Can ...
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2answers
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Is there a general formula for the eccentricity $e$ of ellipsoid?

For an ellipse of eccentricity $e$ the formulas are: ${x^2 \over a^2} + {y^2 \over b^2} = 1 \\ e = \sqrt {1-\left({b \over a} \right)^2}$ what about the "3D case"?
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1answer
143 views

Cylinder between two circles in $\mathbb{R}^3$

Let $\mathbf{p}, \mathbf{q} \in \mathbb{R}^3$, $r_p, r_q \in \mathbb{R}^+$ and $\mathbf{n}_p,\mathbf{n}_q \in S^2$. Further let $\mathbf{n} = \frac{\mathbf{p}-\mathbf{q}}{||\mathbf{p}-\mathbf{q}||}$. ...
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Formula relating three coplanar vectors

Let $\mathbf{\vec{u}}$, $\mathbf{\vec{v}}$ and $\mathbf{\vec{w}}$ be Euclidean vectors in $\mathbb{R}^3$ such that: All of the vectors $\mathbf{\vec{u}}$, $\mathbf{\vec{v}}$ and $\mathbf{\vec{w}}$ ...
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1answer
90 views

Solid angle of a pyramid

Suppose I have a rectangular pyramid. I partition the dihedral angle between a fixed pair of opposite faces into three parts and thereby obtain three sub-pyramids (within the original one). Consider ...
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1answer
34 views

How to get the parametric equation of a rotated cylinder (with certain slope)

I have a basic question but I have failed in solving it. I have the equation of a cylinder which is $y^2 + z^2 = r^2$ (centered in the x-axis). The parametric equation (dependent on $L$ and $s$) is $(...
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2answers
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Prove that volume of a cylinder section cut off by a plane is $\frac16$ of circumscribing parallelepipe

I was given the following question: Given a cylinder circumscribed within a parallelepiped with a square base that has a plane going through the center of the base circle and through one side of the ...
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2answers
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Tetrahedron circumradius in high dimensions [closed]

I guess this answer had already been answered a long time ago, but indeed I cannot find any reference. What is the circumradius of a $n$-dimensional regular hypertetrahedron? Does it approach the ...
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1answer
53 views

Uniform distribution of points on a sphere: only Platonic solids?

I'm quite sure the only way to uniformly distribute $n$ points on the sphere $S^2$ is by inscribing one of the 5 Platonic solids, thus there only exists a solution for $n=4,6,8,12,20$. But am I right? ...
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When can we inscribed sphere in tetrahedron? [closed]

I'm wondering is there any conditions to do that. I believe propably there are but what ?
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1answer
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What is known about Lebesgue's universal covering problem in three dimensions?

Lebesgue's universal covering problem is a relatively well-known open problem in geometry, asking for the convex set of minimal area which contains all planar sets of diameter 1. While the problem is ...
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3answers
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Proving spheres are orthogonal

Given two spheres in $\mathbb{R}^3$: $x^2+y^2+z^2=2ax; \ \ \ x^2+y^2+z^2 = 2by$ and $a,b>0$, and $\gamma$ the intersection of the spheres, show that for any $p_0 \in \gamma$, the spheres are ...
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1answer
28 views

Description of curve arising from rotation of a cube

Take a unit cube, and place it so that one of its body diagonals lies along the $z$-axis. For symmetry, assume that the vertices of the cube are at $(0,0,\pm\sqrt{3}/2)$. Then rotate it about the z-...
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0answers
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Simple solids lifted into four dimensions

Somewhat like the surface of revolution that transforms a line meridian in 2D into 3 space with an extra dimension over its un-rotated form (we rule out an extrusion as a special case of rotation as ...
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1answer
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What's the actual volume of a Great Stellated Dodecahedron? (Im getting seemingly different formulas)

So I was trying to derive the formula for the Great Stellated Dodecahedron starting from a unitary side length Dodecahedron. I managed only to get so far. So when I went to check the actual formula, I ...
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1answer
40 views

Moving Objects in POV-Ray

I´m looking for a way to move objects in POV-Ray. Let´s say, I got $n$ Points in 3D. They are all in one plane. Now I want to create a bodie with the shape of the points as its base and a given hight. ...
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1answer
71 views

Construct any solid with identical plane shapes and the resulting solid must be a fair die?

Take a bunch of identical shapes and stick them together to form a convex 3-d solid. Of course, this won't be possible for most shapes. But when it is possible, one should always get a fair die. All ...
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1answer
45 views

An MC question involing cosine law

I saw this question in an DSE exam. I can solve it via:- Method #1 (Numerical method) (1.1) Letting PQ = 1; (1.2) Turn all lines into numerical data (eg. $QR= \dfrac {1}{ \tan 47^0} = 0.932515$); (1....
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1answer
36 views

How to solve an equation with two lines perpendicular to a plane

Find a plane passing through the line (x − 2)/2 = z + 3, y = 1 and perpendicular to the plane x - 2y + z = 2. How would I go about solving this problem? Would I use two points on the two lines (2,0,-3)...
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1answer
33 views

On unitary transformation for two Irreducible representations of space group 198

I am considering the irreducible representation (IR) of space group 198, which consists of two generators $$ \left\{C_3\right\}: (x,y,z)\rightarrow(z,x,y)\\ \left\{C_{2x}\big|\frac{1}{2},\frac{1}{2},0\...
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2answers
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Rectangular prism with volume and surface area

Here is the question: A rectangular prism has a volume of $720$ cm$^3$ and a surface area of $666$ cm$^2$. If the lengths of all its edges are integers, what is the length of the longest edge? This is ...
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1answer
41 views

The equation of a plane passing through three noncolinear points $p_1 = (x_1 , y_1 , z_1)$, $p_2 = (x_2 , y_2 , z_2)$, $p_3 = (x_3 , y_3 , z_3)$

Show that an equation of a plane passing through three noncolinear points $p_1 = (x_1 , y_1 , z_1)$, $p_2 = (x_2 , y_2 , z_2)$, $p_3 = (x_3 , y_3 , z_3)$ is given by $(p − p_1) \times (p − p_2) \cdot (...
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1answer
27 views

The distance between 2 planes on a cube

Given cube $PQRS-TUVW$. It has the length of the side is $3\sqrt3 cm$. What is the distance between $PRW$ plane and $QVT$ plane? Attempt: Cz there's no information about the labelling, i consider $...
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1answer
38 views

Maximum number of balls included in one ball

I guess this question has received an answer since a long time, but I was not able to find it (bad queries on Internet, I suppose): Take a ball $\mathcal{B}$ of radius $r$ in $\mathbb{R}^3$, for ...
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1answer
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How do I find the lateral area of a cuboid given its height, base area and the area of the diagonal cross-section?

H, M and B are given and I need to find the lateral area (area of all the sides): Sketch of the cuboid Since it's a cuboid, I know that the lateral area is $$S = 2(aH + bH) = 2H (a+b)$$ I found the ...
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1answer
79 views

Define pyramid and cylinder by functions f(x,y,z) >= 0

My thought process: Define the shapes individually For cylinder with radius 0.5 I got the function $(0.5^2-x^2-z^2, 2-y, 0.5-x, 0.5-z, y-0)$ Cylinder $r=0.25$ $0.25^2-x^2-z^2, 2-y, 0.25-x, 0.25-z, y-...
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4answers
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Does the sphere $x^2 + (y-2)^2 + z^2 = 1$ intersect the plane $z-x = 3$? If so, find a point in their intersection. If not explain why.

Does the sphere $x^2 + (y-2)^2 + z^2 = 1$ intersect the plane $z-x = 3$? If so, find a point in their intersection. If not explain why. I feel like I can give a convincing answer with a rough sketch, ...
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2answers
41 views

Find $x+y+z$, where $x, y, z$ are edges of a parallelepiped

A parallelepiped has its edges represented by $x$, $y$ and $z$, these are directly proportional to the numbers $3$, $4$ and $5$ respectively. It is also known that they are, in this order, in ...
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1answer
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Where are the centered trigonal lattices among the 14 Bravais lattices?

I have identified the missing entries in the Bravais lattice table for seven crystal families as the already existing entries int the table. However I'm unable to do so just for the trigonal crystal ...
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1answer
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Calculating integrals with angles (spherical/cylindrical): $r d\phi$ or just $d\phi$?

I'm not sure I understood properly when we need to add "something" before the dVariable. For example, in a practice problem where I was asked to calculate flux out of a cylinder, I ...
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1answer
54 views

Volume of tetrahedron with regard to opposite sides

Prove that the volume of the tetrahedron $ABCD$ is $\frac{1}{6}AB\cdot CD\cdot EF \sin x$ where $EF$ is the shortest distance between $AB$ and $CD,$ and $x$ is the angle between these two lines. I ...
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1answer
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3d Geometry- The base of the pyramid has a trapezoid

The base of the pyramid has a trapezoid whose diagonal is perpendicular to the side, and an $\alpha$ angle with the base. The height of the trapezoid is equal to $h$. Each lateral edge of the pyramid ...
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Comparing CIE-type Color-Spaces

I am not really sure where this question should go because it is about colorimetry, so sorry in advance if I am in the wrong place. I want to start working on a project where I construct a color space ...
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1answer
66 views

Plane intersecting a cylinder: points on that ellipse

I understand that an arbitrary plane intersecting an arbitrary cylinder is described by an ellipse. I want to know how to find the points that lie on that ellipse. Say the cylinder is described by a ...
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1answer
36 views

Write the equation of the line that satisfies the following conditions

Let $\pi$ be the plane with equation $x+y+z=0$, $A$ the point $(2,1,3)$, and $v$ the vector $(1,-2,1)$. Write the equation of the line $r$ passing through the point $B$, symmetric of $A$ with respect ...
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1answer
29 views

Solid defined in 3 dimensions [closed]

Which shape is defined as follows: $A=\{ (x,y,z) \in \Bbb{R}^3 : 0\le{y}\le{1}, 0\le{z}\le{1}, z\le{x}\le{z+1}\} $ I struggle picturing these things.
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1answer
119 views

Four points in space, satisfying the conditions

$A, B, C, D$ are four points in the space and satisfy $\mid \overrightarrow{AB} \mid = 3, \mid \overrightarrow{BC} \mid=7,\mid \overrightarrow{CD} \mid=11$ and $\mid \overrightarrow{DA} \mid=9$. Then $...
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0answers
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Two tubes (infinite right circular cylinders) $A$ and $B$ of equal radii $3$ cross at the angle $90°$ [duplicate]

Two tubes (infinite right circular cylinders) $A$ and $B$ of equal radii $3$ cross at the angle $90°$ (i.e. their axial lines cross at that angle). What is the volume of the intersection of $A$ and $B$...
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2answers
222 views

Parallelepiped and shortest path it can take to the opposite vertex

An ant is sitting in a vertex of a right parallelepiped with edges $2, 3, 12$. What is the length of the shortest path it can take to the opposite vertex? Now, I tried to imagine that and we have a $...
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1answer
228 views

tetrahedron volume given rectangular parallelepiped

Let $A$ be a rectangular parallelepiped with edges of lengths $15, 20, 30$. Let $B$ be a tetrahedron on four non-adjacent vertices of $A$ (i.e no two vertices of $B$ share a common edge of $A$). ...
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1answer
50 views

Proving that midpoints of 4 sides and the incenter in a tetrahedron are coplanar, given some area conditions

I'm having trouble solving this problem, can someone help? In tetrahedron $ABCD$, the sum of the areas of faces $ABC$ and $ABD$ is equal to the sum of the areas of faces $ACD$ and $BCD$. Let $E$, $F$,...
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1answer
47 views

Determining analytically the number of times a line intersects a general 3D surface

Consider a general surface and a line in $\mathbb{R}^3$. Given equations for both the surface and line, is there a way to analytically determine the number of times the line intersects the surface? I ...

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