A sphere of radius #a# is cut in two parts by a plane at a distance of #a/2# from the surface of sphere. What is the ratio between two parts?

1 Answer
Feb 12, 2018

Please see below.

Explanation:

Let us consider the sphere, cut by a plane at a distance of #h# from its surface as shown. The sphere has a radius #a#. However, we take it as #r# and then substitute #r=a#. Observe that domain of #h# is given by #0<=h<=2r#.

https://en.wikipedia.org/wiki/Spherical_cap

The plane cuts the sphere at a distance of #h# as shown. Formulas are available here. It divides sphere in two parts, one the blue cap and the lower pink portion.

Volume of the cap is given by formula #pi/6h(3a^2+h^2)#. Note that #r^2=a^2+(r-h)^2# or #a^2+h^2=2rh#, where #a# is radius of circle formed by plane, when it cuts the sphere. As we are using #a# for radius and #h=a/2#, let us use #r# for radius of this circle.

And we have #r^2+a^2/4=2axxa/2# or #r=sqrt3/2a# and hence volume of cap is

#pi/6xxa/2(3xx(sqrt3/2a)^2+a^2/4)#

= #pi/12a((9a^2)/4+a^2/4)#

= #pi/12xx10/4xxa^3=(5pi)/24a^3#

As the volume of sphere is #4/3pia^3#,

the volume of lower pink portion is

#4/3pia^3-(5pi)/24a^3=(27pi)/24a^3#

and ratio of volumes of two parts is

#(5pi)/24a^3:(27pi)/24a^3#

i.e. #5:27#

A detailed discussion on the topic and deduction of formulas is available here.