Cone (EntityTopic, 11)
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| + | <[#ontology [kind topic] [cats 3D Curved Tapertope] [alt [[freebase:03bzp0]] [[wikipedia:Cone_(geometry)]]]]> | ||
{{STS Shape | {{STS Shape | ||
| - | | image=<[# | + | | image=<[#embed [hash 5YKZM3N2DR4ZGBBREJKXT1AV1Q] [width 180]]> |
| dim=3 | | dim=3 | ||
| - | | elements= | + | | elements=1 [[circle]], 1 conical nap, 1 circular edge, 1 [[point]] |
| genus=0 | | genus=0 | ||
| ssc=(xy)P | | ssc=(xy)P | ||
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}}}} | }}}} | ||
| - | A '''cone''' is a special case of a [[pyramid]] where the base is a [[circle]]. | + | A '''cone''' is a special case of a [[pyramid]] where the base is a [[circle]]. It is bounded by its circular base and a curved surface |
The cone is one of the few [[curved]] [[polyhedron|polyhedra]] that satisfy [[Euler's formula|Euler's F + V = E + 2]]. | The cone is one of the few [[curved]] [[polyhedron|polyhedra]] that satisfy [[Euler's formula|Euler's F + V = E + 2]]. | ||
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*Variables: | *Variables: | ||
<blockquote>''r'' ⇒ radius of base of cone<br> | <blockquote>''r'' ⇒ radius of base of cone<br> | ||
| - | ''h'' ⇒ height of cone</blockquote> | + | ''h'' ⇒ perpendicular height of cone</blockquote> |
*All points (''x'', ''y'', ''z'') that lie on the surface of a cone will satisfy the following equations: | *All points (''x'', ''y'', ''z'') that lie on the surface of a cone will satisfy the following equations: | ||
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*The [[hypervolume]]s of a cone are given by: | *The [[hypervolume]]s of a cone are given by: | ||
<blockquote>total edge length = 2π''r''<br> | <blockquote>total edge length = 2π''r''<br> | ||
| - | surface area = π''r'' | + | surface area = π''r''(''r'' + √(''r''<sup>2</sup> + ''h''<sup>2</sup>))<br> |
| - | volume = π''r''<sup>2</sup>''h'' | + | volume = {{Over|π|3}} · ''r''<sup>2</sup>''h''</blockquote> |
*The [[planar]] [[cross-section]]s (''n'') of a cone are: | *The [[planar]] [[cross-section]]s (''n'') of a cone are: | ||
| - | <blockquote>[!x,!y] ⇒ '' | + | <blockquote>[!x,!y] ⇒ isosceles [[triangle]] of base length 2''r'' and perpendicular height ''h''<br> |
| - | [!z] ⇒ circle of radius (''r'' | + | [!z] ⇒ circle of radius (''r'' − {{Over|''nr''|''h''}})</blockquote> |
== Arrinder == | == Arrinder == | ||
An ''arrinder'' is the [[surface of revolution]] of an [[arrow]], just as a [[cone]] is the surface of revolution of a [[triangle]]. It can also be thought of as a cone with a smaller cone removed from the base. As such, this shape's [[volume]] is the difference between the volume of the two aforementioned cones. | An ''arrinder'' is the [[surface of revolution]] of an [[arrow]], just as a [[cone]] is the surface of revolution of a [[triangle]]. It can also be thought of as a cone with a smaller cone removed from the base. As such, this shape's [[volume]] is the difference between the volume of the two aforementioned cones. | ||
| - | + | {{Clear}} | |
{{Trishapes}} | {{Trishapes}} | ||
{{Tapertope Nav|7|8|9|111<br>Cube|2<sup>1</sup><br>Cone|[11]<sup>1</sup><br>Square pyramid|hedra}} | {{Tapertope Nav|7|8|9|111<br>Cube|2<sup>1</sup><br>Cone|[11]<sup>1</sup><br>Square pyramid|hedra}} | ||
Latest revision as of 14:29, 26 March 2017
A cone is a special case of a pyramid where the base is a circle. It is bounded by its circular base and a curved surface
The cone is one of the few curved polyhedra that satisfy Euler's F + V = E + 2.
Equations
- Variables:
r ⇒ radius of base of cone
h ⇒ perpendicular height of cone
- All points (x, y, z) that lie on the surface of a cone will satisfy the following equations:
Unknown
- All points (x, y, z) that lie on the edges of a cone will satisfy the following equations:
x2 + y2 = r2
z = 0
- The hypervolumes of a cone are given by:
total edge length = 2πr
surface area = πr(r + √(r2 + h2))
volume = π∕3 · r2h
- The planar cross-sections (n) of a cone are:
[!x,!y] ⇒ isosceles triangle of base length 2r and perpendicular height h
[!z] ⇒ circle of radius (r − nr∕h)
Arrinder
An arrinder is the surface of revolution of an arrow, just as a cone is the surface of revolution of a triangle. It can also be thought of as a cone with a smaller cone removed from the base. As such, this shape's volume is the difference between the volume of the two aforementioned cones.
| Notable Trishapes | |
| Regular: | tetrahedron • cube • octahedron • dodecahedron • icosahedron |
| Direct truncates: | tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate |
| Mesotruncates: | stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron |
| Snubs: | snub staurohedron • snub rhodohedron |
| Curved: | sphere • torus • cylinder • cone • frustum • crind |
| 7. 111 Cube | 8. 21 Cone | 9. [11]1 Square pyramid |
| List of tapertopes | ||
