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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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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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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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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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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Reflection of a point about a straight line in 3-D space [closed]

Let there be an arbitrary point $P(-2, 5, 7)$ in 3-D space & a straight line having the equation $$\frac{x-2}{5}=\frac{y-1}{4}=\frac{5-z}{3}$$ What will be the point of reflection about the above ...
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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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35 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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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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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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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
45 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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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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32 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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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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53 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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37 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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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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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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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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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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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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111 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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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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123 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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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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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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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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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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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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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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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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66 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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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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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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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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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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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 $2^3 = 8$ ...