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How do we say "within a kilometer radius spherically"? When we say radius, we think of a circle, so how do you specify that you're thinking of a sphere when you say "within a kilometer radius"?

Sayaman
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  • This is a great question because as an English Language question it has a different context compared to a mathematics question. Thanks! – JBH Jun 08 '19 at 21:31
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    A radius is indifferent of whether its 2D circular or 3D spherical. The context will imply the extra or lesser dimension. –  Jun 09 '19 at 03:59
  • If you communicate with a physicist, you can raise a smile by saying within a 1/300000 second light cone :-) – Jens Jun 09 '19 at 08:44
  • @Jens Light cones are 4D object. Also, they are infinite. Giving a dimension doesn't make sense. – Acccumulation Jun 10 '19 at 15:20

6 Answers6

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One could say "spherically", or "within a one-kilometer sphere". In some contexts the three-dimensionality will be obvious, as when one is speaking ot the relative position of satellites, say, or stars

  • There are ten satellites within a fifty kilometer radius
  • There are fewer than twelve stars within ten light-years of the sun.

But there is no special word or phrase that I know of for use in this situation, perhaps because it doesn't come up very often in ordinary speech.

David Siegel
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    Technically, a radius is spherical in 3D space. We think of radius of a circle if the space is just planar. – rexkogitans Jun 08 '19 at 16:30
  • @rexkogitans There can still be a circle, and indeed a line, in three-dimensional space (and in solid geometry) and a circle still has a radius. – David Siegel Jun 08 '19 at 16:40
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    Also an ellipse or an Ellipsoid has a radius -- in fact it has more than one (along the major and minor axes). – David Siegel Jun 08 '19 at 16:49
  • a circle in 3D space is the set of points that results from intersecting a sphere with a plane. Note that circle and sphere have the same definition. – rexkogitans Jun 08 '19 at 19:19
  • But when I say "there were no other cars to be seen within a 100m radius", it's very unlikely that I looked up and checked the sky. – gnasher729 Jun 08 '19 at 21:19
  • @rexkogitans while you're absolutely correct, the average person thinks more in terms of a map, not 3D space. Ask 100 people on the street to explain what "a 1 km radius" means and the only one that would respond with the word "sphere" would be an airplane pilot. Everybody else is thinking "circle" because that what you do when you drive a car to find your local Starbucks. – JBH Jun 08 '19 at 21:29
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    @DavidSiegel you're right that this is incredibly rare in ordinary speech. Where is the sphere? Is its equator equal to the surface of the ground? Is the sphere wholly above ground? Even when talking about a nuclear blast, we're only talking about a hemisphere. As you've answered, the context of the statement will be the only solution 99.9% of the time. – JBH Jun 08 '19 at 21:34
  • @JBH - Unlike standard nuclear weapons, a neutron bomb will destroy, e.g., all life within a one kilometer radius of the blast in ALL directions. The part that's science fiction is that it actually only penetrates the earth several feet and it will not leave all the buildings standing; all nukes can be NBs if you lower the yield enough. – Mazura Jun 09 '19 at 00:25
  • @Mazura JBH said nothing about neutron bombs and also said "hemisphere", strongly implying that they already know that nuclear weapons don't penetrate the ground much. – David Richerby Jun 09 '19 at 09:23
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    If you only said "within a one-kilometer sphere" the assumption would be diameter, not radius. – sirjonsnow Jun 10 '19 at 12:30
  • @sirjonsnow has put his finger on the problem with "within a one-kilometer sphere", which is that it is not possible to know whether it's 1 km radius or diameter. I suspect that more people would think it's diameter, but I'd also object to calling a 1-km-radius sphere a "two-kilometer sphere", because some would take it the other way. – Monty Harder Jun 10 '19 at 15:05
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Pretty much just like that. "Radius" is used for spheres as well as circles.

... anything inside a sphere of one-kilometer radius around the object.

Alternately, just say

... anything within one kilometer of the object

assuming it's obvious you're talking about spatial rather than surface distances.

Andrew
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You can say:

One kilometer in any direction.

Paresh
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    This is very context-dependent. If you're on Earth and say "one kilometer in any direction", people will think of the circle surrounding them on the Earth's surface; if you're in the International Space Station, people will think of the sphere surrounding them. (If you're sitting on a train, people might even think one kilometer along any railway line.) – David Richerby Jun 09 '19 at 09:25
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I usually say 'within a spherical radius of x from this point'.

In 3d computing and visual effects, this comes up more often than you would think.

openCivilisation
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"Radius" is redundant. The term "within" already conveys the concept. If you want to emphasize that it's three dimensional, you can say that: "Within one kilometer, in any of the three dimensions".

Acccumulation
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We say this:

Within a ten-foot-square area.
Within a ten-foot-cube area.

The equivalent for a sphere, and extending it to a kilometre, would be:

Within a one-kilometre-sphere area.

Note that you can also add a d to the end of all of these (squared, cubed, and sphered). While the version without a d is more an adjective and the version with a d is more a verb, I've heard both square and squared, as well as cube and cubed, and I couldn't say if one is more obviously (and objectively) natural than the other. I simply used the adjectival version for all three of my examples.

Jason Bassford
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    The first two statements are ambiguous as it isn't clear if one is referring to 10 feet as being the length of one side, or the resulting square area or volume. Similarly, "one-kilometre-sphere" is ambiguous as to whether 1 km refers to the sphere's radius, diameter, or volume. And although you provide a link to justify the term "sphered", I have never heard this in the wild. All volumes are expressed in cubic terms, regardless of object shape. So not sure if this answer achieves the precision asked for by the OP. – Michael MacAskill Jun 08 '19 at 08:27
  • @MichaelMacAskill I don't think anybody would misunderstand the meaning of the terms 10x10 room or 10x10x10 room, nor would the use of square or cube here normally be thought of as describing something different than that. While square and cube on their own might mean something else (especially, say, a square deal), context—using them in an adjectival noun phrase—should easily determine the intention. Anybody who isn't familiar with sphered would look to a dictionary anyway. (Note that when you mention a 10x10 room, you aren't concerned with height—or volume.) – Jason Bassford Jun 08 '19 at 10:25
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    Indeed, nobody would misunderstand 10 × 10 room, as there is no ambiguity. But "ten-foot-square" is ambiguous. It can easily be though to be "10 square feet", for example, rather than your intended 100 square feet. But again, none of that is relevant to the OP's question, except inasmuch as "one-kilometre-sphere" is still desperately ambiguous. Perhaps not surpassing, as your post refers twice to areas (of cubes and spheres), rather than volumes. – Michael MacAskill Jun 08 '19 at 10:31
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    Cubes and spheres have a volume, not an area. This can hardly be called nit-picking, seeing as it goes to the heart of the OP's question. – TonyK Jun 08 '19 at 11:47
  • @TonyK A cube and a sphere both have both a volume and an area, a surface area. Buit the OP mwas askign about a distance, whioch isn't really either. – David Siegel Jun 08 '19 at 14:38
  • @Michael MacAskill A "10 squar fot area" is clear. it can be, say 10x1. A "ten-foot square area" is 10x10.(100 square foot, but in that specific shape) Neither is ambiguous, although some people carelessly confuse the latter with the former.. – David Siegel Jun 08 '19 at 14:42
  • "square" and "cube" as verbs most often mean "raising to a power. "Square" often means 'to make square" that is, to force to a right-angled shape, often in such forms as "square up" or "square off". "Cube" as a verb is much rarer. aside from the mathematical sense. "Sphere" as a verb is so rare, i don't recall ever hearing it used, and I would advise any learner not to use it. – David Siegel Jun 08 '19 at 14:47
  • @DavidSiegel: the OP's usage, "within a ten-foot-cube area", makes it clear that it is the volume of the cube that is relevant. (And BTW, cubes and spheres don't have an area; they have a surface area.) – TonyK Jun 08 '19 at 16:48
  • @TonyK The question does not say "within a ten-foot-cube area", It says "within a 10-kilometer radius" which means a distance, not an area or a volume. That can imply either a circle (if on a surface) or a sphere (if considered spatially. And the surface area of a sphere is often called just its area. Google "area of a sphere". – David Siegel Jun 08 '19 at 16:57
  • Sorry but "one-kilometer-sphere area" is just wrong. Spheres most naturally describe volume, and areas are about surfaces. If you wanted to say the surface of a one-kilometer sphere, you'd say "the surface of a one-kilometer sphere." If you meant the volume (which the question does), you wouldn't say "area. – David Richerby Jun 09 '19 at 09:27