The rate at which your ice melts does depend upon it's shape. To be
specific, different shapes have different amounts of surface area. For
example, your cube shaped ice has six squares on it's surface, where as
the half-moon or half-cylinder shaped ice has two semicircles and two
rectangles (one is straight and the other curved). So even though both
ice shapes have the same mass, density and volume, they have different
amounts of surface area, depending upon the dimensions (length, height,
radius) of the shapes. By increasing the surface area, the rate of a
process (such as ice melting) increases as more of the ice is exposed to
the warmer atmosphere.
Volume of a cube = L³
L = length of the cube side
Volume of a half-cylinder = PI r²h/2
r = radius of the cylinder
h = height of the cylinder
Surface area of a cube = 6L²
Surface area of the half-cylinder = PI r² + dh + PI rh
d = diameter of the cylinder (equal to 2r)
The first term is the area of the two semicircles
The second term is area of the flat rectangle
The third term is the area of the curved rectangle
If you put values into these equations, you will find that the half-moon
shape ice has greater surface area. However, the surface area of this
shape can change if you alter some of the dimensions.
In the worked calculations that I have done, I assumed that each ice shape
weighed 8g and that ice has a density of 0.917g per cubic centimetre, so
each ice shape has a volume of about 8.72 cubic centimetres.
As the volume of the cube is L³ = 8.72 cubic centimetres, the cube root of
8.72 is 2.06, so L = 2.06 cm. From this, the surface area was found to be
about 25.46 square centimetres.
For the half-moon (half-cylinder), the volume is PI r²h/2 = 8.72 cubic
centimetres. Assuming the height (or thickness) of the half-moon to be
1cm, r² = 5.55 so by taking the square root of 5.55, the radius may be
found to be 2.36 cm (hence d = 4.72 cm)....