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# If a sphere with radius r is inscribed in a cube with edges of length

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Re: If a sphere with radius r is inscribed in a cube with edges of length [#permalink]
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Bunuel wrote:
AbdurRakib wrote:
If a sphere with radius r is inscribed in a cube with edges of length e, which of the following expresses the relationship between r and e ?

A. $$r=\frac{1}{2}e$$

B. r = e

C. r = 2e

D. $$r=\sqrt{e}$$

E. $$r=\frac{1}{4}e^2$$

A sphere inscribed in a cube has the diameter equal to the edge of the cube. Thus, 2r = s, which is the same as r = s/2.

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Re: If a sphere with radius r is inscribed in a cube with edges of length [#permalink]
Top Contributor
Bunuel wrote:
Bunuel wrote:
AbdurRakib wrote:
If a sphere with radius r is inscribed in a cube with edges of length e, which of the following expresses the relationship between r and e ?

A. $$r=\frac{1}{2}e$$

B. r = e

C. r = 2e

D. $$r=\sqrt{e}$$

E. $$r=\frac{1}{4}e^2$$

A sphere inscribed in a cube has the diameter equal to the edge of the cube. Thus, 2r = s, which is the same as r = s/2.

Check other 3-D Geometry Questions in our Special Questions Directory.

Thank you very much
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If a sphere with radius r is inscribed in a cube with edges of length [#permalink]
Can someone please explain why the edge of the cube must be equal to the diameter of the sphere? Is it because the inscribed sphere must touch all the points on the square . Am i correct??
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Re: If a sphere with radius r is inscribed in a cube with edges of length [#permalink]
longhaul123 wrote:
Can someone please explain why the edge of the cube must be equal to the diameter of the sphere? Is it because the inscribed sphere must touch all the points on the square . Am i correct??

Yes. In geometry "inscribed" means drawing one shape inside another so that it just touches. It does not mean just inside.
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Re: If a sphere with radius r is inscribed in a cube with edges of length [#permalink]
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longhaul123 wrote:
Can someone please explain why the edge of the cube must be equal to the diameter of the sphere? Is it because the inscribed sphere must touch all the points on the square. Am I correct??

Sure the inscribed circle must touch all the four sides of the Rectangle so the side of the rectangle is same as the diameter. see the images attached that also show circumscribed circle, in this case the diagonal of the rectangle serves and the diameter
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dia.png [ 89.93 KiB | Viewed 52812 times ]

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Re: If a sphere with radius r is inscribed in a cube with edges of length [#permalink]
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AbdurRakib wrote:
If a sphere with radius r is inscribed in a cube with edges of length e, which of the following expresses the relationship between r and e ?

A. $$r=\frac{1}{2}e$$

B. r = e

C. r = 2e

D. $$r=\sqrt{e}$$

E. $$r=\frac{1}{4}e^2$$

Since the sphere is inscribed in the cube, the length of an edge of the cube = the diameter of the sphere. Recall that the diameter is twice the radius. Thus, e = 2r, or 1/2e = r.

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Re: If a sphere with radius r is inscribed in a cube with edges of length [#permalink]
AbdurRakib wrote:
If a sphere with radius r is inscribed in a cube with edges of length e, which of the following expresses the relationship between r and e ?

A. $$r=\frac{1}{2}e$$

B. r = e

C. r = 2e

D. $$r=\sqrt{e}$$

E. $$r=\frac{1}{4}e^2$$

First of all, if you have a sphere that is inscribed inside of a cube,

that means that the cube's side is going to be exactly twice the radius.

So the radius is going to be half of the edge, and that's about it.

You don't need to calculate anything. The correct answer choice is A.
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Re: If a sphere with radius r is inscribed in a cube with edges of length [#permalink]
If the sphere is inscribed in a cuboid instead of a cube (i.e., sides are not of equal lengths), is it correct to say that the diameter of the sphere will be equal to the shortest length of the cuboid? Thanks!
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If a sphere with radius r is inscribed in a cube with edges of length [#permalink]
Top Contributor
AbdurRakib wrote:
If a sphere with radius r is inscribed in a cube with edges of length e, which of the following expresses the relationship between r and e?

A. $$r=\frac{1}{2}e$$

B. r = e

C. r = 2e

D. $$r=\sqrt{e}$$

E. $$r=\frac{1}{4}e^2$$

As the radius of the sphere is half of the edge of the cube. The edge of the cube is $$e$$, which equal to the diameter of the sphere. The radius $$r=\frac{1}{2}e$$

The answer is $$A$$.
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Re: If a sphere with radius r is inscribed in a cube with edges of length [#permalink]
AbdurRakib wrote:
If a sphere with radius r is inscribed in a cube with edges of length e, which of the following expresses the relationship between r and e ?

A. $$r=\frac{1}{2}e$$

B. r = e

C. r = 2e

D. $$r=\sqrt{e}$$

E. $$r=\frac{1}{4}e^2$$

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Re: If a sphere with radius r is inscribed in a cube with edges of length [#permalink]
Video solution from Quant Reasoning:
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Re: If a sphere with radius r is inscribed in a cube with edges of length [#permalink]
Sometimes, at the GMAT test, is better draw the figure to visualize the math relations.
This method gives you more confidence in the measurements relations and avoid erros on easy questions like this.
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8AD76C22-DA6A-43CF-86C3-8B183E81DCBB.jpeg [ 158.9 KiB | Viewed 10712 times ]

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Re: If a sphere with radius r is inscribed in a cube with edges of length [#permalink]
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Re: If a sphere with radius r is inscribed in a cube with edges of length [#permalink]
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