# Tagged Questions

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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### Relationship between circumscribed sphere radius and edge length of a dodecahedron? [duplicate]

Hello and I'm a secondary student doing a math exploration, but I'm currently stuck with this problem... Can anyone kind enough to show me the derivation of the relationship between the circumscribed ...
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### Stacking the dual tetrahedron in an ordinary tetrahedron. Is it possible?

I'm thinking about the following problem. Introduction First let me introduce the problem with a 2D example. The area of the triangle constructed by connecting the midpoints of a triangle is 1/4 of ...
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### Spheres and skew lines [closed]

G is a given sphere in the space. For any line e that has no common point with G, define the line f as the conjugate of e with respect to G if f joins the points of tangency on the two planes tangent ...
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### Volume of 3 intersecting cylinders

I have the following problem: Find the volume of the intersection of $3$ cylinders that lie in the plane, each of radius $1$ and with an angle between each pair of cylindrical axes of $\pi/3$. I ...
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### Diagonals of squares on curved functions

I just came across an integration problem. It is very easy to plug numbers into the steps of the solved problem and arrive at the right answer, but I don't understand one of the choices of formulas ...
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### What is the minimum information needed to determine a line in 3D?

Motivation: A Line in $\Bbb R^2$ Any line can be uniquely determined by two points. In $\Bbb R^2$, a point is uniquely determined by two values (its $x$ and $y$ coordinates). Hence, to uniquely ...
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### Can the inscribed angle theorem be generalized to solid angles in 3D? And beyond to n-dimensional space?

The "inscribed angle theorem" is a common 2-dimensionl plane geometry fact. It states that for a circle the angle formed between any two points on the circumference with the center is twice the angle ...
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### What's the maximum number of faces a convex polyhedron can have, given that it's polyhedron with all the same faces?

I know there's a polyhedron named a disdyakis triacontahedron, it has 120 faces and they're all the same. Could there be a polyhedron with a larger number of faces? Can it be arbitrarily large?
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### Find a plane perpendicular to $yz$, passing by a point and making an angle with another plane

The problem is to find the equation of a plane (let's call it $A$) that is perpendicular to the $yz$ plane, containing the point $P(2,1,1)$, and making an angle of $\cos^{-1} \frac{2}{3}$ with the ...
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### Silhouette curves in quasi-axisymmetric geometries

I am trying to find a way to identify silhouette curves in geometries which are largerly axisymmetric but with non-axisymmetric features (e.g. a cylinder with a a few small holes drilled in) and run ...
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### In a pyramid with a square base all edges have the same length. Find the angle between skew medians of two lateral faces.

So, I have square pyramid ABCDE, with E being the vertex. Segment ME is on lateral face AED and segment BP is on lateral face BEC. Segment EZ is perpendicular to the base ABCD of the pyramid. I'm ...
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### Ellipsoid but not quite

I have an ellipsoid centered at the origin. https://en.wikipedia.org/wiki/Ellipsoid Assume $a,b,c$ are expressed in $mm$. Say I want to cover it with a uniform coat/layer which is $d$ mm thick ...
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### Center of mass of a tetrahedron

I need to prove that the medians of a tetrahedron are concurrent, and to find the ratio at which they intersect each other. I cannot use coordinate geometry, and I must use barycentric arguments in ...
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### Surface area and the Volume of a spiked icosahedron by using integration

I am doing my math project on the properties of a spiked icosahedron. I already calculated the surface area and the volume through other methods, but the teacher asked me to find a generic formula for ...
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### A hemisphere is inscribed in a cube

Finding the largest cube inscribed in a hemisphere has been considered here previously. So let's consider the reverse relationship: A hemisphere is inscribed in a cube with an edge of $1m$. What ...
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### Tetrahedron packing in Cube

I'm thinking about following solid geometry problem. Q: Suppose you have a box of "cube" shape with edge length 1. Then, How many regular tetrahedrons(with edge length 1) can be in the box? So, this ...
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### Depth of the Ice Cream in a Cone

I received the following question in maths today and I don't know how to tackle it. "The volume of the ice-cream is half the volume of the cone. The cone has a 3cm radius and height of 14cm. What is ...
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### What is circumradius $R$ of the great disnub dirhombidodecahedron, or Skilling's figure?

The vertices of a uniform polyhedron all lie on a sphere. Out of curiosity, I looked at the circumradius $R$ of the $75$ polyhedra (non-prism) in the list (which assumed side $a=1$). For irrational ...
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### How to derive the volume of a tetrahedron with the following data? [closed]

The vertices of a tetrahedron are:- A - (0, 0, 0) B - (0, 0, a) C - (0, b, 0) D - (c, 0, 0) Prove that the volume is:- 1/6 abc. A figure will be helpful.
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### I have a convex hull with the facets in 3D. How do I compute the volume?

I have constructed a convex hull using Randomized Incremental Algorithm and I have the facets of the same. I need to compute the volume of this hull. Would some please share the algorithm for doing ...
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### Intersection of cone and cylinder layout formula for sheet metal application

A common part in HVAC is a cylindrical pipe intersecting a truncated cone. I am designing a machine to mass produce this part. I would cut the parts out of sheet metal and roll them up to form the ...
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### Volume of a 'cylinder with rounded sides'

I need to find the volume of a torus-shaped object, but it which doesn't have space between the ring. We can find the volume of the ring, but what about the inner part? PS: What is that shape called?...
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### Interesting cube subdivisions: what is going on here, and what are these polytopes?

I was messing around recently with a unit cube. If you draw vertices on the midpoint of each edge of the cube, then connect those points by new edges, you will form the wireframe of what I figured ...
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### Coplanarity of tangent points

If the segments AB, BC, CD and DA are tangent to a sphere, how to prove that the tangent points are coplanar?
In 100 Great Problems of Elementary Mathematics by Dorrie, it is proved that there are only five possible tessellations of the sphere using congruent regular (spherical) polygons: $4$ regular ...