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Does the isosceles triangle in a cone, having as 'legs' the elements of the cone and as its base the diameter of the cone's circular base have its own name?

I am talking about something like this:

depiction

Community
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bp99
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  • A little sketch or picture might help. –  Dec 12 '16 at 20:09
  • True, but @Glorfindel has just posted a perfect illustration below. – bp99 Dec 12 '16 at 20:11
  • Isn't that a triangle? https://en.wikipedia.org/wiki/Cone See picture on right –  Dec 12 '16 at 20:11
  • I know that it's a triangle, I think I may have explained the question wrong. I would like to know whether this specific triangle (of a cone) has its own name. – bp99 Dec 12 '16 at 20:12

2 Answers2

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"Apical cross-section." In other words a planar cross-section through the apex of the cone. Actually, there are three possibilities where the intersecting plane passes through the apex:

· The plane intersects the cone only at its apex. The locus is, of course, just a single point.

· The plane only intersects the cone through one element. The locus is a single straight line.

· The plane intersects the cone in two of its elements. The locus is two intersecting straight lines.

Senex Ægypti Parvi
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It seems that the object you describe is an isosceles triangle. It is derived from Ancient Greek; iso means 'equal' and sceles comes from the Greek word for legs. The triangle has two legs/sides of equal length.

If you vary the 'wideness' of the cone, you can get any isosceles triangle you want. The only special one I can think of is the cone with a right angle, for which the corresponding section is an isosceles right triangle.

enter image description here

source: Frederick Converse Beach Encyclopedia Americana (New York, NY: Americana Company, 1903)

Glorfindel
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  • Exactly, it is, but doesn't it have a specific name? I mean this one, which is so special (having the cone's circular base's diameter as its base and having as its 'legs' the elements of the cone (slant height)) – bp99 Dec 12 '16 at 20:09
  • I don't know of a specific name for it (neither in English nor in my native language (Dutch)). Mathematically speaking, it isn't even a triangle. Conic sections are taken from cones which stretch indefinitely to each side, so your cut (known as a degenerate conic section) is in fact two intersecting lines. – Glorfindel Dec 12 '16 at 20:11
  • Also, note that by varying the width of the cone base, you can vary the width of the base side of the triangle. So the only special feature is that two of its sides have equal length. – Glorfindel Dec 12 '16 at 20:13
  • This one is also called an isosceles right triangle because it has a right angle. Was that the special one you're looking for? – Glorfindel Dec 12 '16 at 20:15
  • I'm aware of isosceles triangles. No, sorry, it wasn't. What I mean is whether there is a name for this isosceles triangle, that we use when talking about cones. For example, the slant height of a cone is called it's element, however, it's also just a simple line segment with no special properties (expect maybe that it connects a point on the base circle with the apex - mind you, the 'tip' of the cone, which is just a point in 3D space also has its own name, apex), however, it has it's own name. Sorry, it might be complete nonsense to you what I'm saying. – bp99 Dec 12 '16 at 20:23
  • @bertalanp99 If you edit your question so it asks exactly what you want it to (perhaps with an image), I'll migrate it to [Math.SE]. – Andrew Leach Dec 12 '16 at 20:24
  • @bertalanp99 the point is, that if you vary the 'wideness' of the cone, you can get any isosceles triangle. The only special one I can think of is the cone with a right angle, for which the corresponding section is an isosceles right triangle as mentioned here. – Glorfindel Dec 12 '16 at 20:27
  • @Glorfindel I am afraid you missed my point, but I am sure it's because of my lack of knowledge of English. – bp99 Dec 12 '16 at 20:46
  • You mean a special name for it because you cut it out of a cone? Just like the term conic section; a parabola is a conic section but most people encounter it first as the graph of $y = x^2$. – Glorfindel Dec 12 '16 at 20:48
  • @Glorfindel No. I mean a special name for it because it's an important intersection of this solid (the cone). It shows its aperture, slant height, dihedral angles, etc. To generalise, I would like to know the term used for intersecting anything with a plane. For example when you want to depict certain things, let's say how our veins work, you often draw a picture where you just a draw the vein as a circle (as if you cut it somewhere with a plane). This way you can easily show the layers of the vein. I am looking for something similar in Maths. – bp99 Dec 12 '16 at 21:46
  • @bertalanp99 OK, so basically you're looking for the inverse term of Solid of revolution / Surface of revolution. According to the latter article, generatrix could be the term you're looking for. – Glorfindel Dec 12 '16 at 21:48
  • @Glorfindel I am afraid not, because that would mean that a triangle can be a generator / generatrix of a cone. – bp99 Dec 12 '16 at 22:08
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    Well, according to the definition, it does. But I'm all out of options right now. – Glorfindel Dec 12 '16 at 22:09