1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Area of sphere inside cube

  1. Mar 28, 2013 #1
    Ok, so a cube and a sphere are both centered at origo. The cube has side lengths L. I need to know the surface area of the part of the sphere that is inside the cube, for all possible r. There will be three different equations, one for [itex]0 < r < L/2[/itex], when the entire sphere is inside the cube, one for [itex]L/2 < r < L/\sqrt(2)[/itex], when six spherical caps of the sphere stick out of the cube sides , and one for [itex]L/\sqrt(2) < r < L\sqrt(3) / 2[/itex], when only the corners of the cube is visible. I have figured out the two first, but not the last.

    Any help would be greatly appreciated!
     
  2. jcsd
  3. Mar 28, 2013 #2

    phinds

    User Avatar
    Gold Member
    2016 Award

    You need to show some work on your own before we know where you are stuck and how to help.
     
  4. Mar 28, 2013 #3
    OK. The reason I did not show my work is because i believe there must be a much simpler way. But her it is.

    If I am not mistaking, the area X of the sphere INSIDE the cube is the total area A of the sphere, minus the area C of the six spherical caps, plus the area of the overlaps E between the caps near each of the 12 edges (because it has been subtracted twice). So [itex]X = A - 6C + 12E[/itex].

    I have found A and C:
    [itex]A = 4\pi R^2[/itex]
    [itex]C = 2\pi R h = 2 \pi R (R-L/2)[/itex]

    Now I only need E, the intersection / overlap between two adjacent spherical caps. In the attached drawing, the colored area is HALF of E. E has been divided in the middle by the arc of a great circle. Finding the colored area have proved more difficult than I expected, though.
     

    Attached Files:

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Area of sphere inside cube
  1. Area of a sphere (Replies: 9)

  2. Area of sphere. (Replies: 8)

  3. Surface Area of a Cube (Replies: 1)

Loading...