1. The problem statement, all variables and given/known data A torus has a major radius and a minor radius. When R>r by a magnitude of at least 4x, it comes to be a slim ring looking shape. When R>r by a magnitude of 1/2, the shape looks to be a donut. When R=r, the torus shape looks more like a sphere except with a small gap in the center of the shape. I want to be able to show that volume and surface of a torus is greater than that of a sphere at the nanoscale level. The purpose of this problem is to show that a torus at the nanoscale level can carry more particles and has a greater surface area, making this shape a more effective method than a spherical liposome. 2. Relevant equations Torus: V= 2*pi^2*R*r^2 SA = 4*pi^2*R*r Sphere: V= (4/3) * pi * r^2 SA = 4 * pi * r^2 3. The attempt at a solution I started first by comparing the V and SA using R=5 and r=2.5 for simplicity and because I want the donut looking shape compared to the small ring (by using R= (1/2)r ) Final numbers for torus: V= 616.225 SA = 492.98 Final numbers for sphere: V = 26.17 SA = 78.5 I then calculated the magnitude of difference in V and SA, which turned out to be 23x and 6.3x respectively. I expected to see the same results of magnitude at the nanoscale level. Because I have to keep in mind the major radius of the torus, I made R+r = 1e-9. For the sphere it was just r = 1e-9. Stating that R = 1/2r, I plugged that in my equation and made (1/2r) + r = 1e-9. r + 2r = 2e-9 3r = 2e-9 r = 6.7 e -10 R = .5 (6.7e-10) <---- R = 1/2r R = 3.35e-10 Figuring out my values, I then used those numbers in my V and SA formulas Torus: V= 2.97 e -29 SA = 8.9 e-18 Sphere: V= 4.2e-18 SA = 1.256e-17 As you can see, the calculations from the first results are very different and at the nanoscale this shows that the volume of a torus is actually less than that of a sphere. This makes sense because I am multiplying by a very small number (R) unlike that in the sphere formula. Did I do my work wrong somewhere? Why is that at the nanoscale level the torus is less effective than the sphere?