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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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Nonperiodic space-filling polyhedra

For periodic space-filling polyhedra, the maximum number of faces seems to be 38, according to On Space Groups and Dirichlet-Voronoi Stereohedra. For non-periodic space-filling polyhedra, the ...
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Why does the formula for the volume of a rhombic triacontahedron contain $\sqrt{5}$?

The volume of a rhombic triacontahedron is calculated by utilising the formula: $$V = 4a^3 \sqrt{ 5+2 \sqrt{5}}$$ where $a$ is the length of an edge. The rhombic triacontahedron is a polyhedron with ...
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What solid does the truncation of an icosahedron approach?

i was watching some old maths fun related stuff on youtube, and i stumbled across this video: https://www.youtube.com/watch?v=cwWBpjeyRS0 in which the guy mentions how a football is obtained by ...
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Algorithm to build a dice with given probabilities

Let's define a dice as a solid that, if rolled over a perfect horizontal plane, ends up being in a physically stable unambiguous state labelled $n$. The dice has $N$ states. Each state $n$ has a ...
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geometry - how to construct the “net” of a two-frequency geodesic dome of radius x?

I'm new to this community. I am trying to construct a small (16 cm radius) geodesic dome with the northern celestial hemisphere inscribed on the exterior surface. I have a few questions, but I'll ...
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3answers
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Neighboring solids in tetrahedral-octahedral honeycomb

In the tetrahedral-octahedral honeycomb, each vertex seems to be incident to 6 octahedra and 8 tetrahedra: Such simple combinatorial fact is probably well-known, or perhaps even obvious. However, ...
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3D broken stick, triangles only.

On mathoverflow is the question Probability that a stick randomly broken in five places can form a tetrahedron. For this question I'm interested in a simpler variant, looking at the triangles only. ...
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Hex map equivalent in 3-dimensional space

In 2-dimensional space, hex maps are great for gaming because any single cell's center is equal distance to any adjacent cell's center, as seen below: I am looking to do a 3-dimensional board game in ...
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2answers
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Octahedrons whose faces are congruent quadrangles

There is a octahedron which the faces are all congruent quadrangles. Let $M$ the set of length of edges of faces of the octahedron. Prove that $|M| \le 3$. Prove that all faces have two ...
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1answer
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How can I check if a sphere collides with a cylinder?

I am working on the following problem: I have a cylinder (of which I know: the radius $r_c$ and points $P_0=(x_0,y_0,z_0), P_1=(x_1,y_1,z_1)$ at the center of the two bases) and a sphere (of which I ...
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What is the name of the shape generated by the vertices of the Soviet pennant on the Luna 2 spacecraft? Is it an polyhedron?

xkcd 2125 titled "Luna 2" refers to the object shown below. All faces seem to be pentagons. (found in this question.) Copy of the Soviet pennant sent on the Luna 2 probe to the moon, at the Kansas ...
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A problem with tetrahedron [closed]

Let ABCD be a tetrahedron with the property that opposite edges are equal. We know that the angle between the planes ABD and BCD is $90^\circ$ and the angle between (BCD) and (CAD) is $60^\circ$. ...
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4 Spheres all touching each other??

If there are 4 spheres all touching each other and 3 of them have diameters 4, 6 and 12 what is the diameter of the fourth one? I imagine it like 3 balls on a flat table touching each other and then ...
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2answers
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Shortest distance between two points on the surface of a closed cylinder

What is the shortest distance between two points on the surface of a closed cylinder? I understand simple euclidean distance will work if both points are on curved surface, but I am looking for a ...
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What is the volume of the cone resulted when the first-quadrant portion of the graph of $y = −2x + 4$ is revolved about the $y$-axis? [closed]

I am having trouble with visualizing this problem: QUESTION: What is the volume of the cone resulted when the first-quadrant portion of the graph of $y = −2x + 4$ is revolved about the $y$-axis? ...
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Medial axis of symmetric solid & volume

I have a problem and would appreciate some pointers. In short, I have a solid which is produced by revolving a 2D polyline around an axis, thus producing a solid of revolution. Now: 1) Is there some ...
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Unfolding a pentacube into a net

I am trying to unfold a J2 pentacube into a flat net (also non-edge intersecting) such that the net fits in the smallest possible rectangular area. So far I have managed to unfold it to a 6 by 7 grid:...
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1answer
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Find Radius of hemisphere inside a square pyramid [closed]

You have a hemisphere that touches all faces of a square pyramid (except the base). You are given the length of each side, as well as the height of the pyramid and asked to find the radius of the ...
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How to calculate solid angle of nonspherical surface?

My objective is to calculate (or find an expression) for the solid angle of a circular loop (parametrized by $(x,y)=(r \cos{x_i}, r \cos{y_i})$, where $0\leq x, y \leq 2\pi$) on the surface defined as:...
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2answers
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Can there be two adjacent solid angles?

Thanks for reading. My real question is the second part - in the first part I'm just explaining myself. Please read through! Thanks. In 2D geometry, it is easy to picture what it means to add up 2 ...
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Maximum (expectation of) volume of a regular tetrahedron whose angles are i.i.d. random variables

I am proposing here a multi-d version of the question I proposed here and from which I have really learnt a lot. Problem. Given three points "on the Earth", i.e. three couples of longitude/latitude $...
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Finding the volume of a right cylinder in terms of full surface area and a variable

Let S be the full surface area of a right cylinder. Let H be the height of the cylinder and r be the radius of it's base. Let m = H-r. Find the volume V of the cylinder in terms of S and m.
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1answer
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Volume of a Steinmetz-like solid

Consider three cylinders intersecting with a unit cube. Their intersection within the unit cube produces a 3-sided solid with a volume of about .386. One cylinder has center axis (0,0,1) to (0,...
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Viewing a circle from different angles - is the result always an ellipse?

Take a piece of rigid cardboard. Draw a perfect circle on it. Hold it up, and take a picture, with the cardboard held perpendicular to the direction we're looking. You get a photo that looks like ...
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1answer
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Volume and surface area of sphere, cone, cylinder etc

Why isn't the volume of a sphere: $\pi$$^\text{2}$$r^\text{3}$, instead it is $\frac{4}{3}$$\pi$$r^\text{3}$? Like wise the surface area is 4$\pi$$r^\text{2}$and not 2$\pi$$^\text{2}$$r^\text{2}$. ...
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The square on the diagonal of a cube has area 1875. Find a side of the cube and its total surface area.

The square on the diagonal of a cube has an area of 1875 cm$^\text{2}$. Find One side of cube The total surface area of the cube Moreover, what does ‘square on the diagonal of a cube’ ...
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Need help to solve a question of Mensuration

The radius of base and slant height of a conical vessel is $3$ cm and $6$ cm respectively. Find the volume of sufficient water in the vessel such that when a sphere of radius $1$ cm is placed into it, ...
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2answers
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What is the accuracy of SVD in 3d transformations

I have a triangle $x$ with points $x_1,x_2,x_3\in\mathbb{R}^3$ that were measured at one location. The triangle was then transformed to $\bar x$ where the points were measured again as $\bar{x}_1,\bar{...
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1answer
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Get direction vector of a line from a point and its rotation.

The title pretty much says what I need. Some details about my problem: I have a cylinder. I have a point outside this cylinder. I want to find the direction from said point to the line that passes ...
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4answers
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Does there exist a tetrahedron, so that every edge is the side of an obtuse angle of a face?

I have the following the question with me: "Does there exist a tetrahedron, so that every edge is the side of an obtuse angle of a face?" The problem looks easy but I am unable to prove it. Any ...
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1answer
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Area of a triangle in space using determinants from 3 points

I know there is a way to find the area in plane using 3 points but when it comes to space(3D) does it work too? if so how should it look? Thanks
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1answer
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to find the slant height of a conical frustum given its top radius, it's lateral surface area and it's common central angle

I've searched high and low for a formula for finding the slant height of a conical frustum given it's top radius, it's lateral surface area and it's common central angle but have come up empty handed. ...
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Polyhedra with coplanar non-adjacent faces

Two non-adjacent faces of a polyhedron are called $\textit{buddies}$ if they lie on the same plane. Call a polyhedron $\textit{nice}$ if every face has a buddy. What is the smallest $\textit{nice}$ ...
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A pyramid $OABC$ has vectors $\vec{OA}=a$, $\vec{OB}=b$ and $\vec{OC}=c$.

A pyramid $OABC$ has vectors $\vec{OA}=a$, $\vec{OB}=b$ and $\vec{OC}=c$. The vectors $v_1,v_2,v_3$ and $v_4$ are perpendicular to each of the faces of and of magnitude equal to the area of the ...
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2answers
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Snub cube's angles

I am trying to build a snub cube. I have made $6$ squares and $32$ equilateral triangles (out of perler beads if you're curious). I am trying to figure out the angles at which I adjoin the squares to ...
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Finding the maximum of sum

Given a right hexagonal pyramid $SA_1A_2…A_6$. $A_1 = (0;2;0)$ $A_2 = (0;0 ;0)$ $S = (0;1;3)$ Let $A_4 = (a_1, a_2, a_3)$. What is maximum of $a_1 + a_2 + a_3$ over all $A_4$?
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Finding the shortest path between two points on the surface of a cube

A cube with vertices $(0,0,0),(0,0,1),(0,1,0),(0,1,1),(1,0,0),(1,0,1),(1,1,0),$ and $(1,1,1)$ has the point $P_{1}$ with vertices $(\frac{1}{2},0,\frac{1}{4})$ and the point $P_{2}$ with vertices $(0,\...
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1answer
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All the 2-d polygons that can be used to form 3-d polyhedrons

The requirement is that we start with a 2-d polygon and using just copies of this polygon, cobble them together into a closed 3-d polyhedron (no gaps). I want to find a way to identify all such 2-d ...
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Construct a Trigonal Trapezahedron from another platonic solid

Looking into the Tetartoid, which is a version of the Dodecahedron where all pentagons are not regular, described in the answer by Aretino here is a way to construct it from a Tetrahedron. Similarly,...
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1answer
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Find the angles between two solids

I have 2 solids (A and B) and I need to find the three angles between their x, y, and z axes. If I calculate the geometrical center of the two solids (Ax, Ay, Az and Bx, By, Bz), is it correct to ...
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3answers
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Finding the minimum volume of a Tetrahedron.

Suppose you have the surface $\xi$ defined in $\mathbb{R}^3$ by the equation: $$ \xi :\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 $$ For $ x \geq 0$ , $ y \geq 0$ and $ z \geq 0$. Now take ...
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Find the volume of the surface.

Find the volume of the solid in $xyz$-plane bounded by $y=x^2,y=2-x^2,z=0$ and $z=y+3$. I have found the answer $\frac {13 \pi} {6}$. Is it correct at all? Please verify it. Thank you very much. I ...
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Angles subtended by an edge in a regular dodecahedron?

If I have a regular dodecahedron and construct lines between the center of the dodecahedron and its vertices. How do I calculate the angle between such lines, subtended by an edge? This picture can ...
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3answers
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Show that $A^2 +B^2+C^2=D^2$ using the following diagram (tetrahedron)

Slicing a corner off a square gives a right-angled triangle, as shown in the diagram below. The lengths of the sides of this triangle are related by Pythagoras’s theorem: $a^ 2 + b^ 2 = c^ 2$ . Show ...
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3answers
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Unrolled intersection of 2 cylinders

I'm writing a macro with Visual Basic for Autocad. My problem is: 1)I have 2 cylinders with orthogonal vertical axis, so they intersect themselves generating a 3D space curve. Cylinders could have ...
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1answer
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Simplest Unhalvable Shape

Consider a connected 3D-printable shape such as the below. It appears that any plane passing through the centroid will divide the shape into more than two pieces. Define a shape with this property ...
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4answers
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Prove that if in a tetrahedron if two pairs of opposite edges are perpendicular then the third pair is also perpendicular.

Prove that if in a tetrahedron if two pairs of opposite edges are perpendicular then the third pair is also perpendicular. Method: $\vec a+ \vec b + \vec c + \vec d +\vec e + \vec f = 0 \tag0$ ...
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Why no higher-genus regular polyhedra?

It seems to be a fact that there are only five bounded connected non-selfintersecting polyhedra with identical regular-polygon faces and congruent vertices (i.e., you can pick a neighborhood of every ...
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Is there any kind of trigonometry analog for solids?

I'm looking for a general method of calculating angles in convex tetrahedra, a 3-dimensional analog of trigonometry. Have someone established such system in a formal way?
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2answers
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Number of Regular tetrahedron from Unit cube

Can we count number of Regular tetrahedrons formed out of Unit cube? If vertices of Unit cube are taken as $(0,0,0,)$, $(0,0,1)$, $(0,1,1)$, $(0,1,0)$, $(1,0,0)$, $(1,0,1)$, $(1,1,1)$ and $(1,1,0)$ ...