Mathematics

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 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…
bryn
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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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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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How to calculate the area of a 3D triangle?

I have coordinates of 3d triangle and I need to calculate its area. I know how to do it in 2D, but don't know how to calculate area in 3d. I have developed data as follows. (119.91227722167969, 122.7717056274414, 39.3568115234375),…
iamgopal
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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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Hexagons are best for tiling 2D space in terms of perimeter vs area. What's best for 3D space?

If you think of the bee-hive problem, you want to make 2D cells that divide the plane of honey into chunks of area while expending the least perimeter (since the perimeter of the cells is what takes up resources/effort). The solution ends up being…
chausies
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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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Determine Circle of Intersection of Plane and Sphere

How can the equation of a circle be determined from the equations of a sphere and a plane which intersect to form the circle? At a minimum, how can the radius and center of the circle be determined? For example, given the plane equation…
Alekxos
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Planar geometry problems solved by thinking out of the plane

What are some examples of a planar geometry problem with an elegant nonplanar solution? I will post one example as an answer .
Dan
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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 this…
Fine
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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 distance d…
Casper LI
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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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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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