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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Why is the volume of a sphere $\frac{4}{3}\pi r^3$?

I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$, but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $\frac{4}{3}$ is so random! How could somebody guess something like this for…
Larry Wang
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Why is the volume of a cone one third of the volume of a cylinder?

The volume of a cone with height $h$ and radius $r$ is $\frac{1}{3} \pi r^2 h$, which is exactly one third the volume of the smallest cylinder that it fits inside. This can be proved easily by considering a cone as a solid of revolution, but I would…
bryn
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What is the equation for a 3D line?

Just like we have the equation $y=mx+b$ for $\mathbb{R}^{2}$, what would be a equation for $\mathbb{R}^{3}$? Thanks.
Ovi
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The Scutoid, a new shape

The scutoid (Nature, Gizmodo, New Scientist, eurekalert) is a newly defined shape found in epithelial cells. It's a 5-prism with a truncated vertex. The g6 format of the graph is KsP`?_HCoW?T . They are apparently a building block for living…
Ed Pegg
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Will 3 lights illuminate any convex solid?

Can 3 lights be placed on the outside of any convex N dimensional solid so that all points on its surface are illuminated?
Angela Pretorius
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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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Two individuals are walking around a cylindrical tower. What is the probability that they can see each other?

It'd be of the greatest interest to have not only a rigorous solution, but also an intuitive insight onto this simple yet very difficult problem: Let there exist some tower which has the shape of a cylinder and whose radius is A. Further, let…
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Is there a dissection proof of the Pythagorean Theorem for tetrahedra?

Of the many nice proofs of the Pythagorean theorem, one large class is the "dissection" proofs, where the sum of the areas of the squares on the two legs is shown to be the same as the area of the square on the hypotenuse. For example: One…
Jim Belk
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How to find the distance between two planes?

The following show you the whole question. Find the distance d bewteen two planes \begin{eqnarray} \\C1:x+y+2z=4 \space \space~~~ \text{and}~~~ \space \space C2:3x+3y+6z=18.\\ \end{eqnarray} Find the other plane $C3\neq C1$ that has the…
Casper
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What is the name of this 3D shape with 12 outer vertices?

Faces: 48 Outside vertices: 12 Other vertices: 14 (I believe)
Ethan Chapman
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Viewing a circle from different angles - is the result always an ellipse?

Take a piece of rigid cardboard. Draw a perfect circle on it. Hold it up, and take a picture, with the cardboard held perpendicular to the direction we're looking. You get a photo that looks like this: Notice: it looks like a perfect circle in…
D.W.
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Hole inside cube with tetrahedrons at corners?

Given is a unit cube with a tetrahedron at each corner, as shown here for one corner out of the $8$ : It is noticed that the tetrahedrons are not disjoint. Because I cannot look through the cube, I have great difficulty imagining whether there is a…
Han de Bruijn
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What shape is a Calippo?

The Calippo™ popsicle has a specific shape, that I would describe as a circle of radius $r$ and a line segment $l$, typically of length $2r$, that's at a distance $h$ from the circle, parallel to the plane the circle is on, with its midpoint on a…
SQB
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Ellipsoid but not quite

I have an ellipsoid centered at the origin. Assume $a,b,c$ are expressed in millimeters. Say I want to cover it with a uniform coat/layer that is $d$ millimeters thick (uniformly). I just realized that in the general case, the new body/solid is not…
peter.petrov
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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 polyhedron with the same number of vertices and faces…
Ed Pegg
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