Mensuration for a Cube of diagonal 219.97 units: Volume, Surface Area, Diagonal and Circumradius

You are given a cube of diagonal 219.97 units. 
So we may compute that the side = diagonal/√3 = 127 units.

Computing the Total Surface Area, Volume and Diagonal of the cube

Side (or length) of the cube = 127 units. 
Total Surface area of the cube = 4 * side2 = 4 * 127 * 127 = 64516 square units.
Volume of the cube = side3 = 127 * 127 * 127 = 2048383 cubic units. 
Length of the diagonal of the cube = (3 * side2)0.5 = 219.97 units  (Extending the Pythagorean theorem to three dimensions)

You might also make these calculations by considering the cube to be a special case of other 3D hexahedrons: an equilateral cuboid or a regular square prism or a square parallelopiped. 

Circumsphere of the Cube

A sphere touching all six vertices of a cube is known as its circumsphere. 
The circumsphere of this cube as a radius equal to (length of the diagonal)/2 = 109.99

Now, if we know that the cube is made of Aluminum and we measure the side in inches and the density of the material is 0.258 pounds per cubic inch, we can calculate the mass of the cube:
Mass of the cube = Density x Volume = 528482.81 pounds. 
All cubes are geometrically similar. 

Let's compare this current cube (let's call it cube A) with a cube B of side 1 units
The scale factor A : B = 127:1
Ratio of surface areas of A : B  = 64516: 1 = 16129 : 1 (this holds true whether you compare total surface area or surface area per side, or any corresponding pairs of surface areas between the cubes)
Ratio of volumes of A:B = 2048383 : 1 = 2048383 : 1

While working with scale factors, square the scale factor and multiply it by the area of the original figure to determine the area of the new figure.
Similarly the ratio of volumes will be the cube of the scale factor - as you're multiplying the scale factor thrice.

What if the current cube is made hollow with a thickness of 2 units?

In that case, the volume of the cubic gap inside the current cube has a size of 125 units. So the volume of the cubic gaps is 125 * 125 * 125 cubic units = 1953125 cubic units.
So the volume of the material used to construct the cube = Volume of the outer cube - volume of the gap = 2048383 - 1953125 = 95258 cubic units.

You may find some more useful notes and information about the cube over here.

To understand more about the geometric features and properties of cubes, formulas related to mensuration and the principles of cubical or octahedral symmetry - you might find it useful to read the properties of a Cube tutorial over here. Many of these concepts are a part of the Grade 9 and 10 Mathematics syllabus of the UK GCSE curriculum, Common Core Standards in the US, ICSE/CBSE/SSC syllabus in India. You may check out our free and printable worksheets for Common Core and GCSE.