Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [solid-geometry]

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)

In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space. Reference: Wikipedia.

Stereometry deals with the measurements of volumes of various solid figures (three-dimensional figures) including pyramids, cylinders, cones, truncated cones, spheres, and prisms.

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A problem of J. E. Littlewood

Many years ago I picked up a little book by J. E. Littlewood and was baffled by part of a question he posed: "Is it possible in 3-space for seven infinite circular cylinders of unit radius each to touch all the others? Seven is the number suggested…
Old John
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Volume of sphere-cube intersection

I am looking for a formula for the volume $V_i(r,L)$ of the intersection between a sphere of radius $r$ and a cube of edge $L$ with the same center. For $r<\frac L 2$, $V_i$ is simply equal to the volume of the sphere: $$V_i(r,L) = \frac 4 3 \pi r^3…
valerio
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Is there a shape that can be wrapped perfectly?

Wrapping presents in the real world always involves overlapping paper (due to folds, etc). Is there any shape that can (theoretically) be wrapped by a rectangular piece of paper without any overlap (the shape and the paper have the same surface…
Nathan Merrill
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Surfaces of constant projected area

Generalizing the well-known variety of plane curves of constant width, I'm wondering about three-dimensional surfaces of constant projected area. Question: If $A$ is a (bounded) subset of $\mathbb R^3$, homeomorphic to a closed ball, such that the…
hmakholm left over Monica
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How to calculate the dimensions of the required 20 regular hexagons and 12 regular pentagons for a sphere of given diameter (the soccer ball issue)

I wish to calculate the dimensions of the required 20 hexagons and 12 pentagons to tesselate a sphere of given diameter (the soccer ball issue) I need the hexagons and pentagons to be plane figures. The sphere of a given dimension can be inscribed…
WILLIAMO
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Dividing an octahedron into two congruent regions

Consider a regular solid octahedron, which includes the interior. Let $r$ be the subset consisting of the top vertex, two vertices adjacent to it but not to each other, and the two edges between them, as in the diagram above. Let $g$ be the…
MJD
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Intersection of a Cone and Sphere

Show that a the cone $xy + yz + xz = 0$ cuts the sphere $x^2 + y^2 + z^2 = r^2$ into two equal circles and find their area. I have been trying to substitute one of the variables, say $z$, from the equation of the cone and putting that into the…
coolbootgeek
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Volume and surface area of a sphere by polyhedral approximation

Exposition: In two dimensions, there is a (are many) straightforward explanation(s) of the fact that the perimeter (i.e. circumference) and area of a circle relate to the radius by $2\pi r$ and $\pi r^2$ respectively. One argument proceeds by…
Eugene Shvarts
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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 that is facing towards you, which can never exceed…
kevin
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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.
shadow10
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Angle between two planes, finding the second plane

I've got a plane given by $x-y+z=0$ and a line given by $r=(0,0,a)+t(1,1,b)$. How do I find another plane that contains the line and intersects the other plane at an angle of $45^\circ$? Is the vector $s=(1,1,b)$ perpendicular to both vectors $n_1$…
neža ribarič
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What is the 3d location of the tip of a triangular pyramid with faces identically angled from vertical but an arbitrarily tilted base face?

How can I find the 3d location of the tip of a pyramid which has a 3 sided base arbitrarily angled relative to the xy plane and remaining three faces with identical anglation relative to the z-axis. The known variables are: The three xyz vertices…
David
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Find the minimum volume of a trirectangular tetrahedron circumscribing a spherical ball

Assume the corner of a room with the floor and two walls, all three planes meeting each other at $90^0$. Say, the point where all three meet is considered the origin O and you have X, Y and Z axes along the intersection of different two planes. Now…
Prags
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Unfolding a pentacube into a net

I am trying to unfold a J2 pentacube into a flat net (also non-edge intersecting) such that the net fits in the smallest possible rectangular area. So far I have managed to unfold it to a 6 by 7 grid: Note: To aid visual interpretation the colours…
Ian Miller
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Polyhedra with at least 3 pentagonal faces

A convex polyhedron has at least three faces which are pentagons. What is the minimum number of faces the polyhedron might have? I have a polyhedron with seven faces but I don't know whether it is possible with six:
user327929
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