perplexus.info :: Shapes : Solid triangles (2)

Solid triangles (2) (Posted on 2005-12-05)

There are three points on the surface of a sphere centered at origin. One has an x coordinate of 0, another has a y coordinate of 0, and the last has a z coordinate of 0.

What is the biggest possible equilateral triangle that can be made using these three points as the corners? How many equilateral triangles of this size are possible?

What if instead of a sphere, it is a regular octahedron centered at origin, with each of its vertices on an x, y, or z axis?

Submitted by Tristan

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Solution: (Hide)

The largest triangle possible is centered on the origin, and its side length is R√3. If the radius is √2, then here is an example of such a triangle:
(0,1,1)
(-1,0,-1)
(1,-1,0)
There are 8 triangles of this size.
For the octahedron, the largest triangle does not go through the origin, and is in fact the same size as each of the octahedron's faces. There are therefore 8 triangles, corresponding to the 8 faces. However, if you decide that the three points are distinct (in other words, triangle ABC is different from BCA), then there are 16 triangles.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re(3): part 1 spoiler	MindRod	2005-12-06 22:46:41
re(2): part 1 spoiler	Charlie	2005-12-06 10:14:33
re(2): part 1 spoiler	Charlie	2005-12-06 09:27:16
re: part 1 spoiler	MindRod	2005-12-05 22:18:29
re: part 1 spoiler	Tristan	2005-12-05 20:44:00
part 2--the octahedron--spoiler.	Charlie	2005-12-05 09:53:27
part 1 spoiler	Charlie	2005-12-05 09:37:06

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