Help with maths question

Question

A sphere just fits inside a cube. What is the surface area of the sphere as a percentage of the surface area of the cube? Round answer to nearest percentage.

Answer 1

Let's see:
    SA_sphere = 4πr² = π(2r)² = πd²
    SA_cube = 6s²             [note: s=d in this problem]
 
Percent:
   (SA_sphere / SA_cube)(100%) = (πd² / 6d²)100%  =   (π/6)100% = 53%

Answer 2

Hi Mojdeh.

 

If a sphere fits just inside a cube, that means the radius of the sphere is equal to half the side length of one of the squares that make up the cube. Take a look at the crude representation below of a sphere inside a cube:

 

[O]

 

If we measure the diameter of the sphere, it will go from edge to edge of the cube. Thus, its radius must be half an edge-length.

 

The SA of a cube with side length s is:

 

SA_cube = 6*s^2

 

The SA of a sphere with radius r is:

 

SA_sphere = 4pi*r^2

 

We know for this particular cube and sphere, that r = 1/2*s

 

Or

 

s = 2r

 

Substituting:

 

SA_cube = 6*(2r)^2

 

= 6*(4r^2)

 

= 24*r^2

 

You want to know what the SA of the sphere is as a percentage of the SA of the cube.

 

You'd like to know, then:

 

SA_sphere = k*SA_cube

 

where k is a constant of proportionality.

 

k = SA_sphere/ SA_cube

 

k = 4*pi*r^2 / 24*r^2

 

k = pi / 6

 

= ~0.524

 

= 52.4%

 

So

 

SA_sphere = 52.4% * SA_cube

 

or, rounded to the nearest percentage:

 

SA_sphere = 52% of SA_cube

 

Hope this helps.