Spherical shell

<h2 id="volume">Volume</h2>
<p>The <a href="/facts/Volume/aeAXCBUd">volume</a> of a spherical shell is the difference between the enclosed volume of the outer sphere and the <a href="/facts/Sphere/jywYLNM2">enclosed volume of the inner sphere</a>:
</p>

V
              
              
                
                =
                
                  
                    
                      4
                      3
                    
                  
                
                π
                
                  R
                  
                    3
                  
                
                −
                
                  
                    
                      4
                      3
                    
                  
                
                π
                
                  r
                  
                    3
                  
                
              
            
            
              
              
                
                =
                
                  
                    
                      4
                      3
                    
                  
                
                π
                
                  
                    (
                  
                
                
                  R
                  
                    3
                  
                
                −
                
                  r
                  
                    3
                  
                
                
                  
                    )
                  
                
              
            
            
              
              
                
                =
                
                  
                    
                      4
                      3
                    
                  
                
                π
                (
                R
                −
                r
                )
                
                  
                    (
                  
                
                
                  R
                  
                    2
                  
                
                +
                R
                r
                +
                
                  r
                  
                    2
                  
                
                
                  
                    )
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}V&={\tfrac {4}{3}}\pi R^{3}-{\tfrac {4}{3}}\pi r^{3}\\[3mu]&={\tfrac {4}{3}}\pi {\bigl (}R^{3}-r^{3}{\bigr )}\\[3mu]&={\tfrac {4}{3}}\pi (R-r){\bigl (}R^{2}+Rr+r^{2}{\bigr )}\end{aligned}}}

<p>where r is the radius of the inner sphere and R is the radius of the outer sphere.
</p>
<h2 id="approximation">Approximation</h2>
<p>An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness t of the shell:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>

V
        ≈
        4
        π
        
          r
          
            2
          
        
        t
        ,
      
    
    {\displaystyle V\approx 4\pi r^{2}t,}

<p>when t is very small compared to r (
  
    
      
        t
        ≪
        r
      
    
    {\displaystyle t\ll r}
  
).         
</p><p>The total surface area of the spherical shell is 
  
    
      
        4
        π
        
          r
          
            2
          
        
      
    
    {\displaystyle 4\pi r^{2}}
  
.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Pressure_vessel/5mGeFbpQ">Spherical pressure vessel</a></li>
<li><a href="/facts/Ball_(mathematics)/f9vpMVMp">Ball</a></li>
<li><a href="/facts/Solid_torus/WWEAGbU7">Solid torus</a></li>
<li><a href="/facts/Bubble_(physics)/99TogqIG">Bubble</a></li>
<li><a href="/facts/Sphere/jywYLNM2">Sphere</a></li>
<li><a href="/facts/Focaloid/Exz67YAT">Focaloid</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Weisstein, Eric W. "Spherical Shell". mathworld.wolfram.com. Wolfram Research, Inc. Archived from the original on 2 August 2016. Retrieved 7 January 2017. <a href="http://mathworld.wolfram.com/SphericalShell.html" target="_blank">http://mathworld.wolfram.com/SphericalShell.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Znamenski, Andrey Varlamov; Lev Aslamazov (2012). A.A. Abrikosov Jr. (ed.). The wonders of physics. Translated by A.A. Abrikosov Jr.; J. Vydryg; D. Znamenski (3rd ed.). Singapore: World Scientific. p. 78. ISBN 978-981-4374-15-6. <a href="978-981-4374-15-6" target="_blank">978-981-4374-15-6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Spherical shell open-in-new

Spherical shell