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July 22nd, 2008, 10:26 PM  #1 
Newbie Joined: Jul 2008 Posts: 3 Thanks: 0  A cube is inscribed inside a sphere
A cube is inscribed inside a sphere, the radius of the sphere is 6. What is the volume of the cube? Answer: 192 * Square root of 3 can someone please tell me how to get the answer? thank you 
July 23rd, 2008, 06:53 AM  #2 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: A cube is inscribed inside a sphere
We need to find the volume of the cube. To do this,we need to know the length of the edges (or, 1 edge, because they're all the same length), so that we can multiply l*l*l (=l^3). To do this: Draw a line form one corner of the cube to the very opposite corner so we're going through the center of the cube. Since the cube has the same origin as the sphere, and the corners are on the sphere, this means we have the diameter of the sphere. So, the length of this line is 12. Now, we're going to forget about that line for a bit, and come back to it later. But first, we'll call the line d Draw a line from one of the same vertices (doesn't matter which) as d to the opposite vertex on one of the same faces. This line goes in two of the same directions as d (so they can be connected across an edge). We'll call this line c. We'll call the length of an edge e. Now we know, by pythagorean theorem, that c^2=e^2+e^2 (= 2e^2) Also, the angle created between c and e to form a triangle with d is a right angle. If you don't believe this, verify by twisting a right triangle along one of it's axes. So, we can use this again d^2 = c^2 + e^2 = 2e^2 + e^2 = 3e^2 since d=12, 144=3e^2 48=e^2 e=4*root(3) So, we have the length of a side, now we cube it to get the volume: 64*3*root3 = 192*root(3) Cheers, Cory And if you need any clarification, I'm glad to help... It's difficult to describe geometry in words. 
July 23rd, 2008, 08:35 AM  #3 
Newbie Joined: Jul 2008 Posts: 3 Thanks: 0  Re: A cube is inscribed inside a sphere
Hmm, it's very clear up until the 3e^2. I don't know where that came from; can you please elaborate more on that?

July 23rd, 2008, 08:37 AM  #4 
Newbie Joined: Jul 2008 Posts: 3 Thanks: 0  Re: A cube is inscribed inside a sphere
Haha sorry! It's so easy 2e^2 + e^2 is 3e^2. LOL didn't see it for a while, but I got it now, thank you very much.

November 7th, 2013, 11:13 AM  #5 
Newbie Joined: Nov 2013 Posts: 1 Thanks: 0  Re: A cube is inscribed inside a sphere
Hi guys. I'm currently working on a similar problem, and I need some help. The question is the same, except the radius of the sphere is 3 inches. Can you please guide me in the right direction?

November 7th, 2013, 05:07 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,485 Thanks: 2041 
The volume is proportional to the cube of the radius.

September 15th, 2014, 08:58 PM  #7  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,932 Thanks: 1127 Math Focus: Elementary mathematics and beyond  Quote:
the cube is √(3) (make a diagram if you are unsure). But we know this distance is actually 6, so we must have √(3)x = 6. x = 6/√(3), which remains the same after being multiplied by 1, so the edge lengths of the cube are 6/√(3). Cubing this gives 24√(3) cubic inches as the volume of the cube.  
September 18th, 2014, 01:13 AM  #8 
Senior Member Joined: Jul 2014 From: भारत Posts: 1,178 Thanks: 230  The radius of the sphere is given, which is equal to the half of the diagonal of the cube. Simply interpret it to be a square inside a circle for a while. As given, the radius = half of the diagonal, so let this be one side of an isosceles triangle formed, since it is a cube, its diagonals and half of the diagonals are equal, so looking at any two adjacent $\frac {1}{2}$ of the diagonals, there is an isosceles triangle formed, whose both equal sides are separately equal to the radius of the sphere and the third side of this triangle is equal to one side of the cube which is required. So, we have the two sides of that isosceles triangle, we find the third side. Finding the Third Side of Isosceles Using the angle (we'll call it theta) opposite the unknown side, you can find its length following this technique: 1. Draw a line from that angle to the midpoint of the unknown side, we'll call it B. This should be perpendicular to that side. 2. You have just formed two right triangles within your isosceles triangle. The hypotenuse of the right angle is your known side, we'll call it A. 3. Your angle theta has now been split in half. Calculate sin(theta/2). 4. Now you have: sin(theta/2) = (B/2)/A [Remember, sine = opposite over hypotenuse.] 5. Rearrange the equation to find B and plug in your numbers: B = 2A*sin(theta/2) Now that we have got the third side of the isosceles triangle, it is equal to one side of the cube, so simply find the volume now by raising this side to power of 3. 
March 3rd, 2019, 01:16 AM  #9 
Newbie Joined: Mar 2019 From: Australia Posts: 6 Thanks: 0  Another question
Question is that a cube that each eight vertices touches the surface of the sphere. Express the side length, s, of the cube in terms of the diameter, D, of the sphere. Any ideas?


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