Yahya Ghasemi Print III pdf

3.3.2.1. Square-cube law
The square-cube law was first described by Galilei in 17
th
century (Galilei and Drake, 1946)
and it defines a mathematical principle which describes the relationship between surface area
and volume related to changes in size. When an object undergoes a proportional increase in 
size, its volume grows faster than its surface area. The effect of square-cube law becomes 
especially significant for calculation of specific surface area of finer particles namely powders
and cement i.e. for a given mass of aggregate, the surface area increases with reducing 
particle size. Figure 3.8 shows the effect of size changes of platonic solids on the ratio 
between surface area and volume.
Figure 3.8. Surface area against volume of the platonic solids and a sphere (see Table 3.1).
Eq. 3.26 can be written in its general form where the ratio of surface area to the volume 
defines the shape:
ܽ
௣௢௟௬
=
෍
ܵܵܣ
௜
.
݉
௜
ܸ
௜
.
ߩ
௦
௡
௜ୀଵ
(3.27)
where SSA
i
/V
i
is the surface area to volume ratio of fraction 
i
and is related to the shape as 
shown in Table 3.1. 
0
50
100
150
200
250
300
350
400
0
100
200
300
400
500
Su
rf
ac
e
 Ar
e
a 
Volume 

---

27
Mix design approaches
Table 3.1: Platonic solids used in the calculation of specific surface area. 
Shape
Surface Area
Volume
SSA/V 
Tetrahedron
ξ
3
ܽ
ଶ
ξ
2
ܽ
ଷ
12
14.697
ܽ
Cube
6
ܽ
ଶ
ܽ
ଷ
6
ܽ
Octahedron
2
ξ
3
ܽ
ଶ
1
3
ξ
2
ܽ
ଷ
7.348
ܽ
Dodecahedron
ට
25 + 10
ξ
5
య
ܽ
ଶ
1
4
(15 + 7
ξ
5)
ܽ
ଷ
2.694
ܽ
Icosahedron
5
ξ
3
ܽ
ଶ
5
12
(3 +
ξ
5)
ܽ
ଷ
3.970
ܽ
Sphere
4
ߨܽ
ଶ
4
ߨܽ
ଷ
3
3
ܽ
Substituting spheres with the platonic solids will not only change the calculated volume and 
surface area but also affects the pace of growth in SSA/Volume ratio according to square-
cube law.