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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Graphing the surface $z = xy$

Let the surface $S \subset \mathbb{R}^3$ be the graph of the function $f:\mathbb{R}^2 \to \mathbb{R}, f (x, y) = xy$. Let $U$ be the portion of $S$ for which $x^2 + y^2 ≤ 2$ and let $C$ be the ...
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70 views

Sketching the surface $x^2+y^2+4z^2 = 1$

Let the surface $S \subset \mathbb{R}^3$ be the solutions of the equation $g(x, y, z)$ $ = 1$ where $g(x,y,z)=x^2 +y^2 +4z^2$. Let $U$ be the finite region of S satisfying $z > 0$ and let $C$ be ...
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1answer
65 views

Does anyone know a book on sketching surfaces?

Is anyone aware of books or sources dedicated to sketching surfaces? Sort of like Forst's An Elementary Treatise on Curve Tracing, but on surfaces; or a book that has a fair few chapters dedicated to ...
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1answer
48 views

Can I observe more than half the surface area of a convex object in one view?

In a court scene in a movie, an eyewitness reported that he had eye contact with "the whole bus" during an event. A lawyer challenged this statement, saying "you can only observe the side of the bus ...
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28 views

$A(1,1,1)$, $B(2,1,2)$, $C(3,2,1)$ and $D(2,3,2)$ form a tetrahedron. If $ABC$ is the base, then what is the height?

$A(1,1,1)$, $B(2,1,2)$, $C(3,2,1)$ and $D(2,3,2)$ form a tetrahedron. If $ABC$ is the base, then what is the height? I found out of the equation of the plane containing A, B and C. It is $$-x + 2y +z ...
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1answer
25 views

Unit cube cut into two parts through its diagonals

Question: A cube having each side of unit length is cut into two parts by a plane through two diagonals of two opposite faces. What is the total surface area of each of these parts? My attempt: ...
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25 views

Volume of a cone [duplicate]

I have a simple question about the formula for the volume of the cone. Let $C$ a cone, which base has radius $r$ and height equal to $h$. So its volume can be compute by the formula: ...
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1answer
31 views

Bounding inequalities in three dimensions

I want to write $z^2 \ge x^2 + y^2$, $x^2 +y^2 +z^2 \le 1$ and $z \ge 0$ in the form $$a \le z \le b, \quad c(z) \le y \le d(z), \quad f(y,z) \le x \le g(y,z)$$ or $$a \le z \le b, \quad c(z) \le ...
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2answers
47 views

How can I solve this line & plane intersect question and verify the given answer? [closed]

Find an equation for the plane that passes through the point $(3,2,1)$ and contains the line of intersection of the planes with equations $x+y+z=3$ and $x+2y+3z=6$. The given answer from the key is: ...
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1answer
59 views

Spheres and skew lines [closed]

G is a given sphere in the space. For any line e that has no common point with G, define the line f as the conjugate of e with respect to G if f joins the points of tangency on the two planes tangent ...
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23 views

Volume of 3 intersecting cylinders

I have the following problem: Find the volume of the intersection of $3$ cylinders that lie in the plane, each of radius $1$ and with an angle between each pair of cylindrical axes of $\pi/3$. I ...
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3answers
189 views

Diagonals of squares on curved functions

I just came across an integration problem. It is very easy to plug numbers into the steps of the solved problem and arrive at the right answer, but I don't understand one of the choices of formulas ...
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2answers
54 views

What is the minimum information needed to determine a line in 3D?

Motivation: A Line in $\Bbb R^2$ Any line can be uniquely determined by two points. In $\Bbb R^2$, a point is uniquely determined by two values (its $x$ and $y$ coordinates). Hence, to uniquely ...
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0answers
59 views

What is the cap body produced by the unit sphere?

For ecach number a > 1, let C(a) be the cap body produced by the unit sphere in E^(3) and the points (+-a,0,0). Calculate the volume, surface area and mean width of C(a). For this question, I don't ...
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0answers
30 views

Can the inscribed angle theorem be generalized to solid angles in 3D? And beyond to n-dimensional space?

The "inscribed angle theorem" is a common 2-dimensionl plane geometry fact. It states that for a circle the angle formed between any two points on the circumference with the center is twice the angle ...
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19 views

The relations between lines in the space

I can't understand why the vertical straight Lines in space are all parallel While the horizontal straight lines aren't parallel
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2answers
60 views

What's the maximum number of faces a convex polyhedron can have, given that it's polyhedron with all the same faces?

I know there's a polyhedron named a disdyakis triacontahedron, it has 120 faces and they're all the same. Could there be a polyhedron with a larger number of faces? Can it be arbitrarily large?
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1answer
26 views

Find a plane perpendicular to $yz$, passing by a point and making an angle with another plane

The problem is to find the equation of a plane (let's call it $A$) that is perpendicular to the $yz$ plane, containing the point $P(2,1,1)$, and making an angle of $\cos^{-1} \frac{2}{3}$ with the ...
2
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1answer
25 views

Silhouette curves in quasi-axisymmetric geometries

I am trying to find a way to identify silhouette curves in geometries which are largerly axisymmetric but with non-axisymmetric features (e.g. a cylinder with a a few small holes drilled in) and run ...
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1answer
31 views

In a pyramid with a square base all edges have the same length. Find the angle between skew medians of two lateral faces.

So, I have square pyramid ABCDE, with E being the vertex. Segment ME is on lateral face AED and segment BP is on lateral face BEC. Segment EZ is perpendicular to the base ABCD of the pyramid. I'm ...
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5answers
804 views

Ellipsoid but not quite

I have an ellipsoid centered at the origin. https://en.wikipedia.org/wiki/Ellipsoid Assume $a,b,c$ are expressed in $mm$. Say I want to cover it with a uniform coat/layer which is $d$ mm thick ...
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1answer
73 views

Center of mass of a tetrahedron

I need to prove that the medians of a tetrahedron are concurrent, and to find the ratio at which they intersect each other. I cannot use coordinate geometry, and I must use barycentric arguments in ...
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0answers
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Surface area and the Volume of a spiked icosahedron by using integration

I am doing my math project on the properties of a spiked icosahedron. I already calculated the surface area and the volume through other methods, but the teacher asked me to find a generic formula for ...
3
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2answers
58 views

A hemisphere is inscribed in a cube

Finding the largest cube inscribed in a hemisphere has been considered here previously. So let's consider the reverse relationship: A hemisphere is inscribed in a cube with an edge of $1m$. What ...
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1answer
76 views

Tetrahedron packing in Cube

I'm thinking about following solid geometry problem. Q: Suppose you have a box of "cube" shape with edge length 1. Then, How many regular tetrahedrons(with edge length 1) can be in the box? So, this ...
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2answers
85 views

Depth of the Ice Cream in a Cone

I received the following question in maths today and I don't know how to tackle it. "The volume of the ice-cream is half the volume of the cone. The cone has a 3cm radius and height of 14cm. What is ...
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3answers
112 views

What is circumradius $R$ of the great disnub dirhombidodecahedron, or Skilling's figure?

The vertices of a uniform polyhedron all lie on a sphere. Out of curiosity, I looked at the circumradius $R$ of the $75$ polyhedra (non-prism) in the list (which assumed side $a=1$). For irrational ...
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1answer
26 views

How to derive the volume of a tetrahedron with the following data? [closed]

The vertices of a tetrahedron are:- A - (0, 0, 0) B - (0, 0, a) C - (0, b, 0) D - (c, 0, 0) Prove that the volume is:- 1/6 abc. A figure will be helpful.
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1answer
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I have a convex hull with the facets in 3D. How do I compute the volume?

I have constructed a convex hull using Randomized Incremental Algorithm and I have the facets of the same. I need to compute the volume of this hull. Would some please share the algorithm for doing ...
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2answers
97 views

Intersection of cone and cylinder layout formula for sheet metal application

A common part in HVAC is a cylindrical pipe intersecting a truncated cone. I am designing a machine to mass produce this part. I would cut the parts out of sheet metal and roll them up to form the ...
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3answers
103 views

Volume of a 'cylinder with rounded sides'

I need to find the volume of a torus-shaped object, but it which doesn't have space between the ring. We can find the volume of the ring, but what about the inner part? PS: What is that shape ...
5
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0answers
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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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23 views

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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1answer
41 views

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
113 views

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 ...
2
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1answer
52 views

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
21 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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1answer
41 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
39 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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1answer
35 views

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
78 views

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
38 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
29 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
44 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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0answers
60 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
54 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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1answer
658 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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0answers
23 views

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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1answer
53 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 ...