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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logarithmic spiral around cone stump

Based on the answer on my previous question I managed to come up with the following equations: $$\begin{eqnarray} k &=& 1 \\ r_\Delta &=& r_b - r_t \\ r(\theta) &=& r_t * ...
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spirals around cone

I have multiple spirals running around a cone. The spirals are $$r_\Delta = r_b - r_t$$ $$x(z) = r_b \cos(z) - r_\Delta z \cos(z)$$ $$y(z) = r_b \sin(z) - r_\Delta z \sin(z)$$ $$d(z) = ...
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Formula of the Radius [closed]

A cyclic quadrilateral has sides equal to 60, 25, 52 and 39 cm respectively. Find the radius of the circle circumscribing the cyclic quadrilateral if the area of the quadrilateral is 1764cm^2
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58 views

How to make an icosahedron from 20 tetrahedra?

To make an icosahedron out of Sierpinsky tetrahedrons is difficult because regular tetrahedra can't tile in space. The dihedral angle of a tetrahedron is ~70.53. So the first step would be to make ...
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23 views

Calculating the volume of a solid of revolution about a line.

A figure is formed by revolving the region bounded by $f(x) = \cos{(x)}$ and $g(x) = \sin{(x)}$ from $0$ to $\dfrac{\pi}{4}$ about the line $y=-1$. This figure is formed by integration of two ...
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45 views

A problem about prism with triangular bases

Consider a prism with triangular base . The total area of the three faces containing a particular given vertex is $k$ . Then is the maximum possible volume of the prism $\sqrt {\dfrac {k^3} {54} } $ ? ...
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32 views

Find corners of a square in a plane in 3d space

I am given two angles (similar to theta and phi in spherical coordinates) from which I can calculate a normal vector to a plane in 3d space. I am also given the center point of the square and the area ...
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30 views

Questions on the relation of the axis of a cone to its conic sections

(1) Does the axis of a cone pass through the foci of any its conic sections? Consider the image below: Is the intersection of the axis of cone and the ellipse the same as the focus of the ellipse? ...
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Volume of intersection between two horn tori

While playing around in Blender, I recently stumbled across a certain shape. The shape is found by taking the volume shared between two identical horn tori rotated at right angles to each other: The ...
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1answer
20 views

Mathematical designation of a cuboid that has one or two edges with infinite length

Is there a mathematical designation for a cuboid that has one or two of its edges with infinite length (in essence, forming an infinite subset of the 3-D space)? This would be some kind of 3-D strip ...
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32 views

Equation of a line that goes through $A(-3,-7,-5)$ and $B(2,3,0)$ and find $C(x, -1, z)$ on the same line

Problem: Find the equation of a line that passes through $A(-3,-7,-5)$ and $B(2,3,0)$ and find $C(x, -1, z)$ on the same line. I have completely forgotten how to solve this and I've been reading ...
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30 views

Using a determinant to find the Cartesian equation for a plane from its parametric equations

This horribly unreadable webpage describes a method to find the Cartesian equation for a plane given its parametric equations. I'll try to type the method out here in a neater fashion: The ...
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1answer
27 views

3D geometry (a bit thoughtful)

When we are actually finding the eqn of a plane parallel to the x axis or any other axis then why do we make the direction ratio of respective axis to be zero ? I mean for example, you represent for ...
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2answers
109 views

Find the spherical angles

I have a problem which I could not resolve. Consider a known single frequency plane wave coming from an arbitrary $(\theta, \phi)$ direction where $(\theta ,\phi)$ are the usual spherical ...
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32 views

Find the minimum total surface area of the cylinder in given circumstances.

Six solid hemispherical balls have to arranged one upon the other vertically .Find the minimum total surface area of the cylinder in which the hemispherical balls can be arranged, if the radii of ...
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14 views

How to find out the interior angle of a regular spherical decagon?

A regular spherical decagon has its each side as a great circle arc of 5 units on a spherical surface with a radius 26 units. How to calculate the value of each interior angle of the decagon? I ...
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1answer
101 views

How to find out percentage of total area covered by 20 identical circles touching one another on a sphere?

20 identical circles touching one anther at total 30 different points (i.e. each one exactly touches three other circles) on a spherical surface with a finite radius. Thus all 20 circles are packed on ...
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2answers
55 views

A-Level/GCSE Geometry textbook? Geometry for STEP and MAT?

everyone. I have been looking for a book that covers the most elementary parts of Geometry, such as similar triangles, circles(arc, sector and others), Pythagorean theorem, Sine and Cosine Laws, so ...
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Twelve identical circles touching one another on the surface of a sphere

Twelve identical circles are to be drawn on a spherical surface having a radius $R$ such that the circles touch one another at 30 different points i.e. each of 12 circles exactly touches other five ...
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44 views

Evaluate the eccentricity of the elliptical section of a right circular cone

A right circular cone, with the apex angle $\alpha=60^{o}$, is thoroughly cut with a smooth plane inclined at an acute angle $\theta=70^{o}$ with its geometrical axis to generate an elliptical section ...
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1answer
43 views

How to find out the solid angle subtended by this slotted section?

I have got to calculate total luminous flux incident on a plane section by calculating the solid angle subtended by the slotted (plane) section, having four identical rectangular slots each having ...
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43 views

How to find out the circumscribed radius of a snub dodecahedron?

A snub dodecahedron is produced by partially rotating all 12 regular pentagonal faces of a small rhombocosidodecahedron. A snub dodecahedron has 80 congruent equilateral triangular faces, 12 congruent ...
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Scalar Triple Product

Prove that if $\:$$\vec{a}$, $\vec{b}$, $\vec{c}$ and $\vec{r}$ are any four vectors and if $[\vec{x} \: \vec{y} \: \vec{z}]$ is Scalar Triple product, Then $$[\vec{a} \: \vec{b} \: ...
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2answers
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What is condition for a convex polyhedron to be uniform?

A uniform polyhedron has all its vertices exactly lying on a spherical surface with a certain radius. Condition: A convex polyhedron will be uniform (i.e. all the vertices will exactly lie on a ...
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1answer
68 views

How to evaluate solid angle subtended by a segmented circle?

The diagram above shows a circular plane, centered at the origin 'O', has a radius $7 cm$. Two identical rectangular strips, each having width $2 cm$, are thoroughly cut off from the circular plane ...
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Uniform Polyhedron with 500 congruent right kite faces!

The diagram above shows a uniform polyhedron having 502 vertices exactly lying on a spherical surface, 1000 edges & 500 congruent right kite faces each having two unequal edges $a$ & $b$ ...
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1answer
47 views

Volume of solid by Cartesian, Cylindrical, & Spherical

I am having trouble just setting up the integrals for this problem. Find the volume of the solid bounded by $x^2 + y^2 = 1, z = 0$, $z = 6$, $y\geq 1/2$. a) Use integration with Cartesian ...
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2answers
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finding a point on a surface? the surface is an ellipsoid

I have drawn the cross-sections of the surface $2(x-1)^2 + (y+2)^2 +z^2 = 2$ for the given planes, but am now asked to write down a point which is on the surface. I have no idea how to go about this, ...
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64 views

Angle between a line and a plane in a cuboid

I was no good at this kind of stuff at school, probably because of bad spacial awareness. I thought I'd revisit it to see if I've improved! I'm stuck on this question: I dropped a perpendicular ...
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67 views

What is the volume of $A'EF-ABD$?

$ABCD-A'B'C'D'$ is a cube with a edge length of $6$. $E$ is the midpoint of $A'B'$ and $F$ is the point on $A'D'$ where $|A'F|=2|D'F|$. The question is: what is the volume of $A'EF-ABD$ ? I ...
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Interception Solid

Could someone draw for me (in $\Bbb R^3$, $x,y,z$ axes) the resultant solid of the interception of $x^2+y^2 \le 1$ and $y^2+z^2 \le 1$? I'm getting a little tired of thinking about the solid but can't ...
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Are knot complements prime 3-manifolds?

Well, that is basically the question. Is the complement of a knot, i.e. an smoothly embedded copy of $S^1$ in $S^3$ a prime $3$-manifold? Here I mean by prime: A connected $3$-manifold $M$ is prime ...
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What is the solid angle at a vertex in a snub cube?

I was able to find or derive an expression for most other Platonic or Archimedean solids, but for the snub cube I was not able to find a value nor find an expression anywhere.
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Finding the length of string wound on a cube.

Consider a cylinder of height $h$ cm and radius $r = \dfrac2π$ cm as shown in the figure (not drawn to scale). A string of a certain length, when wound on its cylindrical surface, starting at point ...
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Finding vertical spacing between string wound on the cylinder.

Consider a cylinder of height $h$ cm and radius $r = \dfrac2π$ cm as shown in the figure (not drawn to scale). A string of a certain length, when wound on its cylindrical surface, starting at point ...
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why aren't prisms archimedian solids?

I don't understand which part of the definition of an archimedian solid excludes prisms from being one. each vertex of a prism has the same polygons around it (4,4,n for a n-gonal prism), and also ...
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Three balls inside a cone

Given a cone with slant height $4$ and radius $2$. Inside there are three identical balls touching each other and lateral surface of the cone. Two of the balls touch the base. What is maximal radius ...
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98 views

How to get projection of ellipsoid onto sphere

I'm trying to get the projection of an ellipsoid onto a sphere. Depicted in the image below, I need the projection of the red ellipsoid onto the unit sphere at the origin. I have tried various ...
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1answer
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Cross Product of vectors in 7 Dimensions [duplicate]

While reading a geometry book I came across something like....... Cross Product is possible only in 3 Dimension system and 7 Dimension system. Why?(or How?)
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perpendicular on a plane

If the plane $lx+my+nz=p$ where $l^2+m^2+n^2=1$ meets the coordinate axes in X, Y, Z and G is the centroid of the triangle XYZ and if the perpendicular to the plane at G, meets the coordinate planes ...
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1answer
18 views

Pyramid Surface Area

Square based pyramid has a side length of $220$ (b) and a height of $105.$ Find the surface area. I tried by "doing" Pythagorean theorem $110^2+105^2=s$ then i did the equation for surface area ...
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What is the topology where all the direct distances are equal to $d_1$ and all the cross distances are equal to $d_2$

What is the topology (2D or 3D representation) that corresponds to the following description: We have $K$ pairs of points, where pair $k$ is denoted as $(P_k,Q_k)$. We suppose that the distance ...
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68 views

Good book for Solid Analytical Geometry?

So my teacher uses this book, William H McCrea's Analytical Geometry of Three Dimensions, but it's awfully hard and dry. I need something with more exercises and better explanations, but that covers ...
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369 views

Concentric Equilateral Triangles

I'm currently researching a particular dynamical system that is very geometric in nature. As part of this, I need to prove the following results (the second obviously implies the first). They are ...
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94 views

Find the height of frustum of a cone

The diagram shows a piece of wood which is obtained by cutting off the lower section of a cone. Using PI = 3.142, calculate the value of x. I seriously have no idea on how to find $x$. Any helps ...
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Mapping Coordinates on a Plane Tangent to a Sphere into Cartesian Coordinates in 3D Space

Before we begin, I must ask you to keep the vocabulary at high school level. These variables define the point the plane needs to be tangent to – the center of the circle is at the origin. r - defines ...
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Length of a right triangle's hypoteneuse projected onto a sphere

Please forgive me if this is the wrong kind of question, but I need someone to verify or refute my work. One leg of a triangle has length, $b$ (base), resulting from angle theta swept out by a ray ...
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Figuring out the major axis and minor axis of a 3D ellipse

If a $3d$ ellipse $S=\{(x,y,z)|(x^2)/(2t) + y^2$ $\le 1, z=t, \frac{1}{2} \le t \le 1$}$ The answer book gives the major axis to be $1$, but shouldn't the major axis be $\sqrt{2t}$? as even the ...
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1answer
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Solid mensuration (circles)

Three circles of different radii are tangent to each other externally. The distance between their centers are $9\ cm$, $8\ cm$, and $11\ cm$. Find the radius of each circle. Find the area in between ...
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Do Meissner bodies which have both rounded edges meeting in a vertex as well as rounded edges surrounding a face minimize volume?

There are known to be two kinds of Meissner bodies constructed from Reuleaux tetrahedra - Meissner body Mv with rounded edges meeting in a vertex, and Meissner body Mf with rounded edges surrounding a ...