A cylinder of radius r sits snugly inside a cube. what expression represents the difference of their lateral

areas?

Lateral area omits the base and top.
For the cube side length = 2r
Lateral area = 4*(4r^2) = 16r^2
For the cylinder
Lateral area = 2pi*r*2r = 4pi*r^2
Difference = (16-4pi)r^2

the area of the square would be the diameter of the circle squared since the diameter of the circle has the same length of a side of the square. Asquare = diameter ^2
the area of the circle is simply pi*r^2
so the difference of their areas would be d^2 - (pi*r^2)

side of cube=2r,
lateral area = perimeter of base x height
=8r x 2r =16r^2

lateral area of cylinder =
circumference x height=
2 pi r x2r =4pi r^2
difference = 16 r^2 - 4pi r^2 =r^2 (16- 4pi)