# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### The Rhombohedron

I am trying to model a rhombohedron (using Blender) as a first pass to building Dürer's solid so I am trying to calculate the (x,y,z) values for a given side length 'a' and angle 'theta' (starting ...
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### Stellating the Octahedron

I have a few related questions and I'd be happy to get some help with any one of them. Is the stellation of a polyhedron generally a 'messy' affair that involves trimming away portions of the ...
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### Calculate the area of a solid of revolution

So the subject title is self-descriptive. My question is how can I calculate the area of a solid of revolution with the information below: ...
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### Prove synthetically: projection of a circle onto a plane is an ellipse

I am wondering how I can prove synthetically that the projection of a circle onto a plane is an ellipse.
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### Measure the slope of a triangle relative to a plane

Suppose I have a list of vertices that form triangles floating in space. How do I measure the slope of each triangle relative to a flat ground plane?
In three-dimnesional space, you are asked to construct a polytope (a geometric object with flat sides) enclosing a volume of $1$. The cost of the model is the sum of the areas of the faces, plus some ...