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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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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138 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
41 views

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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11 views

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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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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28 views

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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85 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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384 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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143 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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2answers
181 views

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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19 views

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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38 views

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
310 views

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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18 views

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 ...
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252 views

Volume of the intersection of two tetrahedra

First, I am far from a mathematician, and this question may be easy, if that's the case, please don't hesitate to let me know. Suppose I have 2 tetrahedra (2 3D simplex), with known ...
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4answers
187 views

Problem of axiomatic euclidean geometry

Let the usual five postulates of Euclid been given. Let's take also this postulate: "If two points lies on the same plane, the whole straight line joining the two points lies on that plane". Is it ...
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1answer
78 views

What do you call 'perpendicular but skew' lines?

For example, the seat tube and rear axle of a bicycle or motorcycle. That is, when viewed from above, the seat tube would appear 'perpendicular' to the rear axle. But in 3d reality, the lines are ...
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64 views

On Euclid's definition of similar and equal solid figures.

The Euclid's definition of similar solid figures is Similar solid figures are those contained by similar planes equal in multitude. And the Euclid's definition of equal solid figures is Equal and ...
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44 views

Determining solid region from bounds of triple integral

If you have an integral such as: $$\int_0^1\int_0^{2-x^2}\int_0^{2-x}f(x,y,z)dydzdx$$ How can you determine the equation for the solid region represented by the bounds of this triple integral? Does ...
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26 views

Sylvester–Gallai theorem in Space

Does this statement true of false? Let $P$ be a set of finite points in space,not all of them are in a plane,and any three points are not a line.Can we always find a plane just pass through three ...
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Proof in a tetrahedron - I added my wrong attempt

please help me solve this problem: There is a tetrahedron (ABCD), where $$ > \angle{ACB}=\angle{ADB}=90^\circ $$ and $$ AC=CD=DB $$ Prove, that $$ AB<2CD $$ My (wrong) attempt: I ...
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1answer
80 views

Feyman's Triangle? How do you find the area of the inner triangle if the outside triangle is equilateral

If triangle $ABC$ is equilateral,$BD/BC=1/3, CE/CA=1/3,$ and $ AF/AB=1/3$. What is the ratio of the area of triangle? I have problems analyzing this triangle I tried to use phythagorean, heron's ...
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1answer
345 views

Ratio of an isosceles triangle to its altitude is 3:4. find the measures of the angles

The ratio of an isosceles triangle to its altitude is 3:4. find the measures of the angles of triangle? So, I have the altitude formula where $h$= $\sqrt s^2-(\frac{s}{2})^2$ so I was thinking when I ...
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3answers
62 views

Find the diameter of the new sphere assuming that the volume of a sphere is proportional to the cube of its diameter

Three spheres of diameters 2,3&4 cm's respectively formed into a single sphere.Find the diameter of the new sphere assuming that the volume of a sphere is proportional to the cube of its diameter ...
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35 views

Find the diameter of the new sphere assuming that the volume of a sphere is proportional to the cube of its diameter [duplicate]

Find the diameter of the new sphere assuming that the volume of a sphere is proportional to the cube of its diameter. I know that diameter is equal to the twice of radius. How can you possibly solve ...
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32 views

How to derive this parametrization of a solid cone

From Wikipedia: A right circular cone with height $h$ and aperture $2\theta$, whose axis is the $z$ coordinate axis and whose apex is the origin, is described parametrically as $$F(s,t,u) = ...
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71 views

Do the $2^n$ hyper-octants of a $n$-sphere always have a $n$-dimensional right angle? Is $\pi/2$ only fundamental in $2$ and $3$ dimensions?

In $2$ dimensions, a $2$-sphere can be divided into $2^2 = 4$ congruent pieces, the $4$ quadrants, each of angle $\pi/2$ radians. In $3$ dimensions, a $3$-sphere can be divided into $2^3 = 8$ ...
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1answer
222 views

Solid mensuration,Solid Geometry book

Hi I'm an engineering student and i really need your help.....can you please suggest some good books you know regarding Solid mensuration/Solid Geometry good for self studying.
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342 views

How to show that five points in ℝ³ are cospherical?

There are many conditions equivalent to the cocircularity of four points on a plane, however i could not find any such lists for the three-dimensional analog. When do five points in three-dimensional ...
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522 views

Volume, Lateral Area, and Surface Area of an Elliptic Conical Frustum

What are the formulae for the volume, surface area, and lateral area (i.e. the surface area without the bases) for the above illustrated elliptic conical frustum? I think I've got the volume figured ...
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1answer
893 views

Surface area of quarter of a Sphere

A quarter sphere with a radius of $10 \text{ units}$. Please help, also remember the sides. I used the normal formula of the total surface area of a sphere and divided it by $4$, then added half the ...
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24 views

How To Find a Set of Points Farthest Apart Within 3D Solid

I am trying to find out a method to solve the following problem: There are two parameters: 1) There is a solid 3D region plotted in a cartesian coordinate system. 2) There is a number of points that ...
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166 views

Proving that the volume of a pyramid is one-third that of its corresponding prism.

Is there any way to prove that for any isosceles triangle, the volume of a solid created when that triangle is projected to a point determining the height above the angle opposite the hypotenuse is ...
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Parametrization of a bounded solid.

So, I have a solid bounded by $z=\sqrt{x^2+y^2}, z=\sqrt{1-x^2-y^2}, z=2$ I had to parametrize it using spherical coordinates so I used $$\begin{cases} x(\rho, \theta, ...
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Determining diameter of strands of twisted rope

Twisted ropes are made by taking 3 strands of smaller rope and twisting them together tightly, by coiling the strands in the same direction, as opposed to braided rope, which requires strands to be ...
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103 views

Volume of Cavity between intersecting multiple Spheres

I want find an equation for this problem: Problem Statement:: I have different size sphere, for example say $R_1$ for Red balls and $R_{2}$ for white Balls, overlapping each other. 1.) I want to ...
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257 views

Why isn't the volume formula for a cone $\pi r^2h$?

So I understand that the volume formula of a cone is: $\frac{1}{3}\pi r^2h$, and when I read about how to derive this formula, it makes sense to me. Funny thing is, I'm stuck on why it ISN'T $\pi ...
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Inscribed spheres in irregular tetrahedra

Let $ABCD$ be an irregular tetrahedron, and let $K$ be the center of its inscribed sphere. Let $M$ be the center of the inscribed sphere of $KBCD$. Are $A$, $M$, $K$ necessarily collinear? I have ...
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62 views

Numerical Integration Over Two Regions of an Ellipsoid

I would like to perform a numerical integration over the surface of an ellipsoid $D$. The domain must be split in two by a plane intersecting the ellipsoid (the intersection is arbitrary), so that we ...
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Pyramid problem - find the height of the pyramid by given edges and a side from the base [closed]

The base of a pyramid is a right triangle with the longest side = 12cm. All non base edges are also equal to 12cm. And how can I find the height of the pyramid?
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1answer
80 views

A cross-section of a pyramid

Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon. This is a repost from this. Can someone help me? Thanks in advance.
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1answer
44 views

Computing Solid Geometry from Surfaces - CAD

I'm an engineer and I recently took a course at my local university about CAD curves and surfaces. I understand how NURBS surfaces work and how to generate the complex surfaces that models consist of. ...
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63 views

Intersection of a plane and a surface of revolution

I'm am stuck on the following problem: I have the equation of a curve in the plane $(x,z)$: $z=f(x)$. I build a surface of revolution in the space rotating this curve around the $z$ axis. I need to ...
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1answer
89 views

Regular pyramid, centre of circumscribed ball=centre of inscribed ball

Prove that if the sum of plane angles in the vertex of a regular pyramid equals $180 ^{\circ}$, then in this pyramid the centers of the inscribed and circumscribed balls are equal. Could you help me ...
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1answer
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Find radius of sphere

Imagine eight spheres of radius 1 that are at $(\pm1,\pm1,\pm1)$. Place sphere A with its center at the origin externally tangent to all of the other spheres. Then place sphere B externally tangent to ...
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Why are polyhedra related to the prime numbers 2, 3 and 5, but not to the prime number 7?

Just take a quick glance at all the numbers in these Wikipedia pages on polyhedra: http://en.wikipedia.org/wiki/Platonic_solid http://en.wikipedia.org/wiki/Archimedean_solid ...
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34 views

volume swept by a triangle

I have two triangles defined in space, say $(a,b,c)$ and $(a',b',c')$, where the second one is the outcome of a linear transform of the first. The lines connecting corresponding vertices ($(a-a')$, ...
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96 views

Solid Trigonometry

V ABCD is a pyramid with the vertex V situated perpendicularly above the centre of the square base ABCD. If $\theta$ is the angle between the edge VA and the base, and $\phi$ is the angle between the ...
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Platonic solids and charged particles

It is known that there are five Platonic solids: If, lets say, there are 4 particles with the same electricity charge and whose movement is constrained to be on a sphere, resulting forces will ...