On-line Calculator for the Areas
of Simple Particle Shapes

Return to Particle Characterization

NOTE: This page uses Javascript to do the calculations. 1. Some Web browsers may not have this capability. 2. If your security settings block scripts you must      allow scripts to run or the calculator won't work.

1. Begin by entering values for the three independent variables in the white windows below (or use the default values shown there). Cube Edge μm L/B for the needle calculation = Note: For a plate, the value of L/B is less than one. Crystal Density kg/m3 See Nelson Table 5-1 for a list of crystal densities. Note: The input and output values for these online calculations are displayed in the commonly-used multiples [μm, ng, m2/g] of the fundamental Systéme Internationale (SI) units [m, kg]. This online calculator converts the display values to and from the fundamental SI units so that the values used in the computations are all in the fundamental SI units.

2. To do a calculation click on => [If it doesn't work, read the red box above.]

3. View the computed results (below):   Spheresamevol Cube Needle/Platesamevol Units Length: D E L For a plate L < B B μm Ratios D/E = 1.2405 E/E = 1 L/E B/E   --- Area μm2 A / Acube 0.8061 1 --- Specific Area m2/g The following parameters do not depend on particle shape as long as the particle is non-porous and is made of homogeneous material (all the same density). : Particle Volume μm3 Particle Mass ng Number of particles per specified mass # / mg

4. Repeat 1-3 (or quit)

UNITS: The equations below are designed for use with the basic System International (SI) units as shown in the table at right.

Variable Symbol SI Unit surface area A m2 needle breadth B m diameter D m edge length E m needle length L m volume V m3 density kg/m3

Formulas for the Area and Volume of a Cube The cube has the simplest formulas for area and volume of any flat-faced solid. If E is the length of the edge of a cube, then the total area of the six square faces is (1.1)       Acube = 6 E2 and the volume of the cube is (1.2)       Vcube = E3 so the surface-to-volume ratio is (1.3)       (A/V)cube = 6 / E

Formulas for the Area and Volume of a Sphere The sphere has the smallest ratio of surface area to volume (A/V) of any shape. If D is the diameter of a sphere, then the area of the sphere is (2.1)       Asphere = D2 and the volume of the sphere is (2.2)       Vsphere = ( / 6) D3 so the surface-to-volume ratio is (2.3)       (A/V)sphere = 6 / D

Formulas for the Area and Volume of a Needle or Plate The simplest formulas for area and volume of any needle or plate shape are for an square prism -- two square ends B on a side, and four rectangular sides B by L. Note: The following equations hold for square prisms with L/B > 1 (needles) and also for square prisms with L/B <1 (platelets). For simplicity we shall use the word needle as the subscript. The total area of the two square sides and the four rectangular sides is (3.1)       Aneedle = 2 B2 + 4 B L = (2 + 4 L/B) B2 and the volume of the needle is (3.2)       Vneedle = B2L, so the surface-to-volume ratio is (3.3)       (A/V)needle = (2 + 4 L/B) / L Note that if L/B=1 we have a cube and these equations reduce to those for the cube.

Question: How do E and A for a cube compare to D and A for a sphere of the same volume? A. Since the volumes of the two shapes are the same we can find the relation between E and Dspheresamevol by setting the equation for the volume of a cube (1.2) equal to the equation for the volume of a sphere (2.2): (4.1)       Vcube = E3 = Vspheresamevol = ( / 6) Dspheresamevol3 Drop the V terms (first and third) to leave (4.2)       E3 = ( / 6) Dspheresamevol3 Solve for Dspheresamevol/E to get (4.3)       Dspheresamevol / E = (6 / )1/3 B. In the equation for the area of a cube (1.1) replace E with right side of Eq. (4.3) to get (4.4)       Acube = 6 (/6)2/3 Dspheresamevol2 and rearrange the right side to get (4.5)       Acube = 61/3 2/3 Dspheresamevol2 C. Divide the equation for the area of a sphere of the same volume(2.2) by the left and right sides of (4.5): (4.6)       Aspheresamevol / Acube = [ Dspheresamevol2] / [(61/3 2/3 Dspheresamevol2] Cancel out Dspheresamevol2 and rearrange to get (4.7)       Aspheresamevol / Acube = (6 / )-1/3 Note: The right side of (4.7) is the reciprocal (1 / the value) of the right side of (4.3). D. We can reduce the right sides of (4.3) and (4.7) to single numbers to get (4.8)       Dspheresamevol / E = 1.2406 (4.9)       Aspheresamevol / Acube = 0.8061 Answer: Compared with a cube, a sphere of the same volume requires a slit 24% wider to pass through than the cube (optimally oriented) and has about 19% less surface area than the cube.

Question: How do E and A for a cube compare to B and A for a needle of the same volume? A. Since the volumes of the two shapes are the same we can find the relation between E and Bneedlesamevol -- with L/B as the shape parameter -- by setting the equation for the volume of a needle (3.2) equal to the equation for the volume of a cube (1.2): (5.1)       Vnsv = Bnsv2Lnsv = Bnsv3 (Lnsv/Bnsv) = Vcube = E3 Note: For simplicity we use the subscript "nsv" to stand for "needle of the same volume". Drop the first, second, and fourth terms to leave (5.2)       Bnsv3 (Lnsv/Bnsv) = E3 Solve for Bnsv / E to get (5.3)       Bnsv / E = (Lnsv/Bnsv)-1/3 where the length-to-width ratio (L/B) is a parameter that affects the area. B. Substitute the right side of (5.3) for B in the equation for the area of a needle (3.1) to get (5.4)       Aneedle = (2 + 4 Lnsv/Bnsv) (Lnsv/Bnsv)-2/3 E2 C. Now divide the left and right sides of (5.4) by the equation for the area of a cube of the same volume (1.1). (5.5)       Ansv / Acube = [(2 + 4 Lnsv/Bnsv) (Lnsv/Bnsv)-2/3 E2] / [6 E2] Cancel out E2 to get (5.6)       Ansv / Acube = (2 + 4 Lnsv/Bnsv) (Lnsv/Bnsv)-2/3 / 6 Note that if L/B = 1 we have a cube and the right side of the above equation reduces to unity (1). D. The first term in () in (5.6) is greater than unity and dominates for L/B > 1; the second term in () is greater than unity and dominates for L/B < 1. See the table below for the value of the right side of (5.6) as L/B changes by a factor of 100. L/B 0.1 0.2 0.33 0.5 1 2 3 5 10 kshape 1.857 1.365 1.159 1.058 1.000 1.050 1.122 1.254 1.508 Answer: Compared with a cube, a needle (or plate) of the same volume has somewhat more surface area, with the relative area becoming larger as L/S moves away from unity (either larger or smaller than 1).