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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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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13 views

How to find the dimensions of a cone given the net

I was wondering if there was a formula for the dimensions of a cone given the sector that is its net. There are a lot of formulas for doing this in reverse, but I can't find any for this. This is ...
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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
19 views

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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39 views

How to derive the volume of a sphere?

I was trying to derive the formula for finding the volume of a sphere. I tried to do it in a same manner as inspired by this answer-Where am I wrong at deriving the formula of volume of cone? ...
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1answer
32 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
34 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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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. ...
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6answers
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Understanding what a plane is in $\mathbb R^3$

I understand how spheres circles and so on work, My interpretation comes from the sum of their co-ordinates equals the $radius^2$. I understand how this works but with planes Im really confused. The ...
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2answers
30 views

Equation used to represent a disc galaxy

I'm trying to create a solid which looks something like a disc galaxy: Key features are: Bulge in the middle Tapered "width" as it extends to a disc shape The end goal would be to use Python to ...
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1answer
23 views

Compute the angle between a line and a plane if the line forms the angles of 45 degrees and 60 degrees with two perpendicular lines lying in the plane

Compute the angle between a line and a plane if the line forms the angles of 45 degrees and 60 degrees with two perpendicular lines lying in the plane. I have no idea how to solve this exercise. I ...
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1answer
30 views

Shortest path to the apex of a cone

This is something I thought about today but have no idea how to approach. We are given a right circular cone with lateral length L and angle at the base $\alpha$. A curve along the surface of the ...
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53 views

Volume generated by revolving $f(x)=\sqrt{x}$ about $g(x)=x$

Recently, I have been trying to refresh my memory about the methods of finding the volumes of solids using integration. I know it's possible to find the volume of a solid generated by revolving ...
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2answers
27 views

What is the ratio of empty to filled volume of the glass?

The base diameter of a glass is $20$% smaller than the diameter at the rim. The glass is filled to half of the height. Then what is the ratio of empty to filled volume of the glass ?
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551 views

Any other Caltrops?

This question has been edited. The regular tetrahedron is a caltrop. When it lands on a face, one vertex points straight up, ready to jab the foot of anyone stepping on it. Define a caltrop as a ...
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Making a Net from a 2D Image

I'm trying to find the volume of the illustration Fig.1, I've taken a scale reference from the medium diameter of a strawberry and I’ve applied this scale to the remaining sides of the shape. Fig.1 is ...
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29 views

Permutations to divide a solid

I have a 3-dimensional cuboid, with dimensions 2x2x1. I wish to divide this into smaller EQUAL sized cuboids of size 2x1x1. Then I want to extend this case for larger cuboids, assuming that equal ...
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51 views

Find equation of plane given plane point

So I am given one plane point $M(5,2,0)$, also two points which are not plane points: $P(6,1,-1)$ distance to plane $1$, also point $Q(0,5,4)$ distance to plane $3$. How find equation of plane with ...
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Quasi-planar bifurcations - What constraints are required when generating the 3 angles?

I have a physical bifurcation where the parent vessel (V$_{12}$) is in the same plane as one of the daughter vessels (V$_{23}$). The other daughter vessel (V$_{24}$) is out of plane to both. $(x_1, ...
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1answer
56 views

In the tetrahedron what is $XY$?

In tetrahedron $ABCD$, $AB = 4$, $CD = 7$, and $AC = AD = BC = BD = 5$. Let $I_A, I_B, I_C,$ and $I_D$ denote the incenters of the faces opposite vertices $A, B, C,$ and $D,$ respecitvely. It is ...
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25 views

Solid Geometry. Finding an angle

$\mathrm{AB}=10$ and it intersects plane a. $\mathrm{A}$ and $\mathrm {B}$ are $2\mathrm{m}$ and $3\mathrm{m}$ far from the plane and the question is to find angle formed by plane a and $\mathrm{AB}$. ...
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To find the volume of a certain solid cone

A solid cone is obtained by connecting (with a line segment in $3$ dimensional Euclidean space ) every point of a plane region $S$ with a vertex not in the plane $S$ . Let $A$ denote the area of $S$ ...
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Volume of an octagonal dome

I need a way to find the volume of an octagonal dome. Can someone help me with the formula for that? Here's an example of the object I am trying to analyze
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what are the symmetries and flags of tetrahedron?

I know that |rot(tetrahedron) | = 12 ( i know how we came up with this number ) my question what is the number of symmetries in tetrahedron ? is it 12 or 24? if is it 24 can anyone explain to me how ...
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Visualizing $180^\circ$ rotational symmetries of a tetrahedron

I am trying to learn about the symmetries of a regular tetrahedron. I understand the identity and all eight $120^\circ$ rotations that keep one vertex fixed, ...
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32 views

Is equal curved surface areas a coincidence

If we take a cylinder of height $2x$ and radius $x$, as well as a sphere of radius $x$, we notice that they have the same curved surface area. Also, if we take the frustum of a cone such that it has ...
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Problem regarding generators through principal elliptic section of $\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$

The question is the following: Show that the generators through any one of the ends of an equiconjugate diameter of the prinicpal elliptic section of the hyperboloid $$\frac{x^2}{a^2} + ...
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point inclusion in a half-plane 3D

I have a 3D half-plane defined using a line segment an a point (as shown in picture taken from here). I am wondering how I can detect if a point belongs to the half-plane. Is there any way to ...
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55 views

Solid of Revolution Problem (semi-circular Groove turned into a Cylinder)

A groove, semi-circular in section and 1cm deep, is turned in a solid cylindrical shaft of diameter 6cm. Find the volume of material removed and the surface area of the groove. The problem is ...
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Arithmetic: mensuration

Image of problem In part three, the question asked for the surface area of the trough, which the formula should be the area of two segments with the addition of the area of the curved side of the ...
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The planes $\lambda x+y+z=\lambda-1,x+\lambda y+z=\lambda-1,x+y+\lambda z=\lambda-1$ form a triangular prism.

The planes $\lambda x+y+z=\lambda-1,x+\lambda y+z=\lambda-1,x+y+\lambda z=\lambda-1$ form a triangular prism.Then find the value of $\lambda $.And find the distance of the origin from an edge of the ...
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1answer
34 views

Let $OABC$ be a tetrahedron with edges and lateral edges given

Let $OABC$ be a tetrahedron with edges $BC=\sqrt2,CA=\sqrt3,AB=2,OA=\sqrt5,OB=\sqrt6$ and $OC=\sqrt7$.Let $G$ be the centroid of triangle $ABC$ and $M$ and $N$ be the midpoints of $OB$ and $CA$.Find ...
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What is the maximum volume of an equilateral triangular prism inscribed in a sphere of radius 2?

What is the maximum volume of an equilateral triangular prism inscribed in a sphere of radius 2? Since the volume of an equilateral triangular prism is $\frac{\sqrt3}{4}a^2h$,where $a$ is the side ...
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A cone with diameter $12$cm and height $8$cm. Find the volume of the inscribed Sphere.

A cone with diameter $12\ cm$ and height $8\ cm$. Find the volume of the inscribed Sphere. Can someone help me solve this maths problem?
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uncertain point in textbook's solution of a “distance from point to line” problem

We have a cube with each side equaling $1$ unit of length. We need to determine the distance from $B$ to the line $A_1C_1$ In my calculation, each side of the yellowish-shaded triangle equals ...
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Lateral area of oblique cylinder and cone

As following picture : I can find the lateral Area of right cylinder and cone. There spread forms are a rectangles and a circular sector. That's very easy. But in the oblique cases ? Are there same ...
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1answer
37 views

What is the intersective curve between sphere and a right cone?

I am confused this picture : What the curve is? I think that the curve is not circle and not the ellipse too, What is the intersective curve?
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Construct the intersection of a cube by a plane through $3$ points on its edges, no pair of which is on the same face

So this is a rather old problem, but I still cannot find a pure constructive solution to it. Please, do not offer me to write a plane equation, etc. I would be grateful, if you offer a solution by ...
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1answer
102 views

Given a solid angle, how do I calculate the surface area subtended by the angle on the surface of a sphere?

I am interested in calculating the elemental surfaces $dS$ (AF) and d$S'$ (EG), given that the solid angle of the very tiny cones' apex, $H$ are both dΩ. Knowing this will help me prove Newton's ...
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Finding the volume of oil in a cylinder which is lying parallel to the ground.

A cylinder of height 2 m and radius 2 m is partially filled with oil. When the cylinder is lying parallel to the ground the height of oil level is 1.5 m from ground. What is the volume of oil? How ...
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Dihedral angle Finding

I meet a wall while some problem solving. The wall is following question. There is a triangle ABC, The vertice A touch bottom plane and the distance from B, C to the bottom : BE = b, CD = c . When ...
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Integer Tetrahedra

The points $\{(0, 0, 0), (12, 27, 44), (48, 0, 20), (48, 0, -64)\}$ have the property that All vertices are on the integer grid, All edge lengths are integers and different $\{51, 52, 53, ...
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Tangent Sphere centers

For circles of Descartes, the "curvatures" or "bends" (1/radius) of four mutually tangent circles have the property that $$2 (c_1^2 + c_2^2+ c_3^2+ c_4^2) = (c_1 + c_2 + c_3+ c_4)^2$$ With the ...
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Coordinates of a rhombohedron

I have found this question (Coordinates for vertices of the "silver" rhombohedron.) which asks: "The "silver" rhombohedron (a.k.a the trigonal trapezohedron) is a three-dimensional object ...
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Ratio between subsegments of space diagonal of a cube

Let $ABCDEFGH$ be a cube where $ABCD$ is the ground face and $E$ is about $A$, $F$ above $B$, and so on. Consider the space diagonal $AG$, and call $P$ the orthogonal projection of $B$ on $AG$. The ...
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Pyramid with unit sides inside a cube

Let $ABCDEFGH$ be a unit cube with base $ABCD$. Let $P$ be the top of the pyramid with base $ABCD$ and all edges of length $1$. One has a standard 2-dimensional projection of this cube on the back ...
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588 views

Finding the base radius of the cone.

A solid cone has a lateral surface area of $100\pi$ square centimeters and a total surface area of $269\pi$ square centimeters. Find the base radius of the cone. I would need to find the radius or ...