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Table 1 Surface area of objects which the circumscribed sphere (one that touches the polygon at all vertices) has a unitary radius (R=1). The number of vertices N and the number of triangular faces N’ are also shown

From: Implementation and clinical application of a deformation method for fast simulation of biological tissue formed by fibers and fluid

Object

N

N’

Analytical Equation

Analytical Result

\( {\displaystyle \sum_{i=1}^N\left|{\overrightarrow{S}}_i\right|} \)

\( {\displaystyle \sum_{i=1}^{N^{\prime }}\left|{\overrightarrow{A}}_i\right|} \)

Octahedron

6

8

\( 4\sqrt{3}{R}^2 \)

6.92820

6.92820

6.92820

Icosahedron

12

20

\( {\scriptscriptstyle \frac{40\sqrt{3}}{5+\sqrt{5}}}{R}^2 \)

9.57454

9.57454

9.57454

≈ Sphere

642

1280

≈ 4πR 2

12.5664

12.5037

12.5065