Y_ jNnB*

{l + A² + tan²*}^8/2’

where R — radius of a sphere in which P may be inscribed,

and

(!-&)}

The method presented seems to have an easier derivation of

the formula than that based on the Biot-Savart Law of mag­netostatics.

2. The article under consideration shows an approach to com­puting the volumes of the regular polyhedra. It is based on the geometric notion of solid angle. The author considers a regular convex polyhedron P, insribed in a sphere of radius Д. Using simple geometric techniques, he finds the formula which expresses the volume of the polyhedron V = Nnlhw/6, where N is the number of congruent faces of P, each with n sides, w is the distance from the center of each face to the center of the sphere; l — the length of each side of a face; h is the apothem. Then finding the constraints to eliminate the quantities /, h and w, the author determines the volume in terms of only jV, n and R. Surprisingly, it turns out that the volume of a dodecahedron inscribed in the unit sphere is larger than that of an icosahedron, which has more faces.

3. The article investigates the problem of computing the vol­umes of the regular polyhedra (also known as Platonic). The author describes an approach where sophisticated physics is replaced by vector calculus. He states that such an approach is geometric and thus conceptually transparent.

The concept of solid three-dimensional angle is presented and three constraints are considered to calculate the volume of the polyhedron. Those three constraints suggested provide with no help of elementary algebra expressions for calculating volumes in terms of only Д, N and n, which are respectively the radius of a sphere containing the polyhedron, the number of its congruent faces and the number of its sides. A sketch of calculations is also given.

2. This article discusses a certain approach to the problem of computing the volume of the Platonic polyhedra. For this purpose the concept of solid angle, that is a natural gener­alization of planar angle, is introduced. It is stated, that with the help of the divergence theorem, one can see, that £1 = f J_K ~gdA, where — the solid angle, and If is a sur­

face in R³, that intersects each ray from the origin in at most one other point. After these notes, a regular polyhedron P is introduced, and w denotes the distance from the center of each face to the center of the sphere, in which P is inscribed, h is the perpendicular distance from the center of a face to a side and l is the length of each side of a face. Abo, P is known to have N congruent faces, each with n sides, and the sphere radius is R. If we have all these data, we can easily find the volume of P: V = Nnlhw/6. The problem is to eliminate /, h, w from this expression, and so one has to find three constraints for these variables. The Pythagorean the­orem, simple trigonometry and the solid angle allow to find

that, finally, V .

2. Several approaches to the problem of computing the volumes of the regular polyhedra are known. Among them there is one based on the Biot-Savart Law of magnetostatics, which was introduced by Paul Shutler and then simplified and developed through replacement of physics with raw vector calculus by Jeffrey Nunemacher.

First, the concept of solid angle is discussed. Its relation to a specific kind of surface integral is established. Second, obvious geometric statements are applied and the volume is expressed in terms of three unknown quantities. To find them constraints are imposed. At this point the first result is ob­tained. At last, the volume is expressed using the well-known algebraic techniques:

_r fiVn^Atan^)

where A =

² {£(!-&)}

{l + A’+tan²*}⁸⁷²’

Besides, an interesting numerical result is provided by the author.

Task 2. Consider the summaries following the article below. De­cide which are acceptable. Then discuss the characteristics of an effective summary and the steps important in writing it.