My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
July 22nd, 2008, 11:26 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
abowlofrice is offline  
 
July 23rd, 2008, 07:53 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.
cknapp is offline  
July 23rd, 2008, 09:35 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?
abowlofrice is offline  
July 23rd, 2008, 09:37 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.
abowlofrice is offline  
November 7th, 2013, 12:13 PM   #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?
_jacey is offline  
November 7th, 2013, 06:07 PM   #6
Global Moderator
 
Joined: Dec 2006

Posts: 21,124
Thanks: 2332

The volume is proportional to the cube of the radius.
skipjack is offline  
September 15th, 2014, 09:58 PM   #7
Global Moderator
 
greg1313's Avatar
 
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,981
Thanks: 1163

Math Focus: Elementary mathematics and beyond
Quote:
Originally Posted by _jacey View Post
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?
Let the edges of the cube measure 1 (for now). So the distance between opposite corners of
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.
greg1313 is offline  
September 18th, 2014, 02:13 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.
Prakhar is offline  
March 3rd, 2019, 02:16 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?
chickensoup is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
cube, inscribed, inside, sphere



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
circle inscribed inside parabolic sector galactus Calculus 5 September 3rd, 2015 02:45 PM
Area of sphere inside cube morjo Algebra 2 March 27th, 2013 03:09 PM
Sphere inscribed in pyramid Daro666 Algebra 0 October 24th, 2012 01:11 PM
Sphere inscribed in pyramid aleschio Algebra 0 September 21st, 2012 01:09 PM
Diameter of inscribed sphere croatianboy Algebra 2 June 4th, 2012 09:45 PM





Copyright © 2019 My Math Forum. All rights reserved.