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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1answer
23 views

3D Geometry concurrency problem

$ABCD$ is a tetrahedron. Let $K$ be the center of the incircle of $CBD$. Let $M$ be the center of the incircle of $ABD$. Let $L$ be the centroid of $DAC$. Let $N$ be the centroid of $BAC$. Suppose $...
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Derive equation for shear modulus $G=E/(1+2v)$

shear modulus, G young's modulus, E and Poisson's ratio, $v$: $G=E/(1+2v)$ I have always wondered how this relation is derived, but have never found a derivation that I could follow online. I ...
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3-D Geometry Problem. Find a curve which touches the straight line.

If two perpendicular tangent planes to paraboloid $x^{2}+y^{2}=2z$ internsects in a straight line in the plane $x=0$, obtain the curve to which the straight line touches. I don't know how to ...
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What is the way to derive the equation of director sphere of any central conicoid?

I am following a book on analytical solid geometry. The book defines the director sphere of a central conicoid as the locus of a point which lies on the intersection of three mutually perpendicular ...
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How do I solve this solid geometry Question?

A Regular tetrahedron is inscribed in a hemisphere of diameter 12 metres such that its apex coincides with the centre of the hemisphere and base fits inside. A flagpole of length 4√6 metres is placed ...
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Is there a common name for solids where $V = h \cdot A_t = h \cdot A_b$

I'm trying to find a name which describes all solids which have these properties: $h = height$ $A_{t} = top\ area$ $A_{b} = base\ area$ $A_{t} = A_{b}$ $V = h \cdot A_t = h \cdot A_b$ Examples ...
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Find the volume(solid) , transform rectangular function to polar function

if want find volume of this problem Under $ f(x)=x^2+y^2-4 $ and inside $ x^2 + y^2=9$ in plane $z=0$ Can I use this integration in polar functions? $$\int_0^{2\pi} \int_2^{3} (r^2 - 4) r dr\,d\...
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A regular tetrahedron is centered at the origin with vertices $ (0,0,1$) and $(a,0,b)$. [closed]

All of its vertices satisfy $x^2+y^2+z^2=1$ Find the remaining two vertices with respect to $a$ and $b$.
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Does a cylinder with equal height and diameter have a special name?

I'm working on z-calibration part for my 3d-printer and I'm wondering if this has a special name? cylinder({r: 5, h: 10}) Basically a cylinder that has a height ...
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99 views

Polyhedra with at least 3 pentagonal faces

A convex polyhedron has at least three faces which are pentagons. What is the minimum number of faces the polyhedron might have? I have a polyhedron with seven faces but I don't know whether it is ...
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70 views

Graphing the surface $z = xy$

Let the surface $S \subset \mathbb{R}^3$ be the graph of the function $f:\mathbb{R}^2 \to \mathbb{R}, f (x, y) = xy$. Let $U$ be the portion of $S$ for which $x^2 + y^2 ≤ 2$ and let $C$ be the ...
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Sketching the surface $x^2+y^2+4z^2 = 1$

Let the surface $S \subset \mathbb{R}^3$ be the solutions of the equation $g(x, y, z)$ $ = 1$ where $g(x,y,z)=x^2 +y^2 +4z^2$. Let $U$ be the finite region of S satisfying $z > 0$ and let $C$ be ...
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1answer
72 views

Does anyone know a book on sketching surfaces?

Is anyone aware of books or sources dedicated to sketching surfaces? Sort of like Forst's An Elementary Treatise on Curve Tracing, but on surfaces; or a book that has a fair few chapters dedicated to ...
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Can I observe more than half the surface area of a convex object in one view?

In a court scene in a movie, an eyewitness reported that he had eye contact with "the whole bus" during an event. A lawyer challenged this statement, saying "you can only observe the side of the bus ...
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$A(1,1,1)$, $B(2,1,2)$, $C(3,2,1)$ and $D(2,3,2)$ form a tetrahedron. If $ABC$ is the base, then what is the height?

$A(1,1,1)$, $B(2,1,2)$, $C(3,2,1)$ and $D(2,3,2)$ form a tetrahedron. If $ABC$ is the base, then what is the height? I found out of the equation of the plane containing A, B and C. It is $$-x + 2y +z ...
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1answer
34 views

Unit cube cut into two parts through its diagonals

Question: A cube having each side of unit length is cut into two parts by a plane through two diagonals of two opposite faces. What is the total surface area of each of these parts? My attempt: ...
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Volume of a cone [duplicate]

I have a simple question about the formula for the volume of the cone. Let $C$ a cone, which base has radius $r$ and height equal to $h$. So its volume can be compute by the formula: $$\text{Vol}(C)=\...
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1answer
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Bounding inequalities in three dimensions

I want to write $z^2 \ge x^2 + y^2$, $x^2 +y^2 +z^2 \le 1$ and $z \ge 0$ in the form $$a \le z \le b, \quad c(z) \le y \le d(z), \quad f(y,z) \le x \le g(y,z)$$ or $$a \le z \le b, \quad c(z) \le ...
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2answers
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How can I solve this line & plane intersect question and verify the given answer? [closed]

Find an equation for the plane that passes through the point $(3,2,1)$ and contains the line of intersection of the planes with equations $x+y+z=3$ and $x+2y+3z=6$. The given answer from the key is: $...
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1answer
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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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3answers
61 views

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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37 views

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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2answers
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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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1answer
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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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1answer
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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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2answers
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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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1answer
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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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1answer
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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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2answers
131 views

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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3answers
107 views

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?
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Proving that there are only five Platonic solids using spherical geometry

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 ...
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1answer
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Which of the $43,380$ possible nets for a dodecahedron is the narrowest?

I want to fit multiple regular dodecahedron nets on to an infinitely long roll of paper. I want this to result in the largest possible dodecahedrons, for a roll of a given width. My hunch is that the ...
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1answer
71 views

Line equation through point, parallel to plane and intersecting line

Write the equations of the line that passes through point $M(1,0,7)$, is parallel with the plane $3x-y+2z-15=0$ and intersects line $\frac{x-1}{4}=\frac{y-3}{2}=\frac{z}{1}$ Alright, so from what I ...
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1answer
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Plane equation through point and parallel to 2 lines

We have the point $A(1,2,1)$ and the lines: d1: $$x+2y-z+1 = 0 , x-y+z-1=0$$ d2: $$2x-y+z=0, x-y+z=0$$ Write the equation of the plane that passes through $A$ and is parallel to the two lines. I'm ...
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45 views

The limit of infinite truncations?

When a regular polyhedron is made to undergo repeated truncations, is there a solid that acts as a kind of limit for this iterated process? That is, say a cube is truncated N times. As N gets larger ...
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1answer
42 views

Why average area of the horizontal slices of the conical frustum doesn't work for it's volume?

I would like to react to one of the answers on this thread (I don't have enough rep to make a comment): Use cylinder's formula for frustum (conical frustum) Where is answered: Essentially, ...
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1answer
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Area of the triangle ABC is $\frac{r^5}{2fgh}$

Through a point P(f,g,h) a plane is drawn at right angles to OP where 'O' is the origin, to meet the coordinate axes in A,B,C.Prove that the area of the triangle ABC is $\frac{r^5}{2fgh}$ where OP=r. ...