Polyhedrons

Excerpts

• Polyhedrons

  A polyhedron is a solid bounded by any number of plane rectilineal figures which are called its faces.

• Polyhedrons

  It will be seen that the demonstration establishes something more than the enunciation states; for it is not assumed that the faces are equilateral and equiangular and all equal.

• Polyhedrons

  If $\mathrm{S}$ be the number of solid angles in any polyhedron, $\mathrm{F}$ the number of its faces, $\mathrm{E}$ the number of its edges, then $\mathrm{S+F=E+2}$.

• Polyhedrons

  Take any point within the polyhedron as centre, and describe a sphere of radius $r$, and draw straight lines from the centre to each of the angular points of the polyhedron;

• Polyhedrons

  but $n$ cannot be less than 3, so that $\dfrac{1}{n}$ cannot be greater than $\dfrac{1}{3}$,

• Polyhedrons

  Thus for a *regular* tetrahedron we have $144\hspace{3pt}V^2=2a^6$.

• Polyhedrons

  A regular octahedron is inscribed in a cube so that the corners of the octahedron are at the centres of the faces of the cube: shew that the volume of the cube is six times that of the octahedron.

• Polyhedrons

  Or we may adopt Carnot’s method, in which this relation is established independently, and the expression for the volume of a tetrahedron is deduced from it;

Equations

• Polyhedrons

  \mathrm{S+F=E+2}

  In any polyhedron, the number of solid angles plus the number of faces equals the number of edges plus two.

• Polyhedrons

  r^2\{s-(m-2)\pi\}

  The area of a spherical polygon with m sides and angle sum s, on a sphere of radius r, is r squared times s minus (m-2) pi.

• Polyhedrons

  4\pi r^2

  The surface area of a sphere of radius r is 4 pi r squared.

• Polyhedrons

  2(S-2)\pi

  The sum of all the plane angles forming the solid angles of any polyhedron is 2(S-2) pi.

• Polyhedrons

  mF=nS=2E

  For a regular polyhedron, the total number of plane angles counted by faces (m F), by solid angles (n S), or by edges (2E) is the same.

• Polyhedrons

  S = \frac{4 m}{2(m+n)-mn}

  The number of solid angles of a regular polyhedron is 4m divided by 2(m+n) minus mn.

• Polyhedrons

  E = \frac{2mn}{2(m+n)-mn}

  The number of edges of a regular polyhedron is 2mn divided by 2(m+n) minus mn.

• Polyhedrons

  F = \frac{4 n}{2(m+n)-mn}

  The number of faces of a regular polyhedron is 4n divided by 2(m+n) minus mn.

• Polyhedrons

  \frac{1}{m}+\frac{1}{n} \text{ must be greater than } \frac{1}{2}

  For a regular polyhedron to exist, 1/m + 1/n must be greater than 1/2.

• Polyhedrons

  \sin{\frac{I}{2}}=\frac{\cos \dfrac{\pi}{n}} {\sin{\dfrac{\pi}{m}}}

  The sine of half the inclination of two adjacent faces of a regular polyhedron equals cos(pi/n) divided by sin(pi/m).

• Polyhedrons

  CE = AE\cot{ACE} = \frac{a}{2}\cot{\frac{\pi}{m}}

  The perpendicular CE from the centre of a face to the midpoint E of an edge equals half the edge times the cotangent of pi/m.

• Polyhedrons

  r = CE\tan{CEO} = CE\tan{\frac{I}{2}} = \frac{a}{2}\cot{\frac{\pi}{m}}\tan{\frac{I}{2}}

  The radius r of the sphere inscribed in a regular polyhedron equals (a/2) cot(pi/m) tan(I/2).

• Polyhedrons

  r = R\cos{aOc} = R\cot{eca}\cot{eac} = R\cot{\frac{\pi}{m}} \cot{\frac{\pi}{n}}

  The inscribed-sphere radius r equals the circumscribed-sphere radius R times cot(pi/m) times cot(pi/n).

• Polyhedrons

  R = r\tan{\frac{\pi}{m}} \tan{\frac{\pi}{n}} = \frac{a}{2} \tan{\frac{I}{2}} \tan{\frac{\pi}{n}}

  The circumscribed-sphere radius R of a regular polyhedron equals r tan(pi/m) tan(pi/n), or (a/2) tan(I/2) tan(pi/n).

• Polyhedrons

  \dfrac{ma^2}{4}\cot{\dfrac{\pi}{m}}

  The area of one face of a regular polyhedron with m sides and edge a is (m a^2 /4) cot(pi/m).

• Polyhedrons

  \dfrac{mFa^2}{4}\cot{\dfrac{\pi}{m}}

  The surface of a regular polyhedron equals (m F a^2/4) cot(pi/m).

• Polyhedrons

  \dfrac{mFra^2}{12}\cot{\dfrac{\pi}{m}}

  The volume of a regular polyhedron equals (m F r a^2 /12) cot(pi/m), with r the radius of the inscribed sphere.

• Polyhedrons

  = abc\surd{( 1 - \cos^2{\alpha} - \cos^2{\beta} - \cos^2{\gamma} + 2\cos{\alpha} \cos{\beta} \cos{\gamma} )}

  The volume of a parallelepiped with edges a, b, c from one vertex and inclinations alpha, beta, gamma is abc times the square root of 1 minus the squared cosines plus twice the product of the cosines.

• Polyhedrons

  OD^2 = a^2+b^2+c^2 + 2ab\cos{\gamma} + 2bc\cos{\alpha} + 2ca\cos{\beta}

  The squared length of the diagonal OD of a parallelepiped equals the sum of the squares of the three edges plus twice each product of two edges times the cosine of their inclination.

• Polyhedrons

  144\hspace{3pt}V^2=2a^6

  For a regular tetrahedron of edge a, 144 V squared equals 2 a to the sixth power.

• Polyhedrons

  144\hspace{3pt}V^2 = -a'^2 b'^2 c'^2 + a^2a'^2 (b'^2+c'^2-a'^2) + b^2b'^2 (c'^2+a'^2-b'^2) + c^2c'^2 (a'^2+b'^2-c'^2) - a'^2 (a^2-b^2) (a^2-c^2) - b'^2 (b^2-c^2) (b^2-a^2) - c'^2 (c^2-a^2) (c^2-b^2)

  The volume of a tetrahedron is expressed in terms of its six edges by this relation, with a, b, c the three edges at one vertex and a', b', c' the opposite edges.

• Polyhedrons

  \cos{ADB}=\dfrac{a'^2+b'^2-c^2}{2a'b'}

  In triangle ADB, the cosine of the angle at D equals (a'^2 + b'^2 - c^2) divided by 2a'b', where c is the side AB.

• Polyhedrons

  1=\cos^2{ADB}+\cos^2{BDC}+\cos^2{CDA}-2\cos{ADB}\cos{BDC}\cos{CDA}

  For four points in a plane with D joined to A, B, C, the squared cosines of the three angles at D, minus twice their product, equal 1.

• Polyhedrons

  0=-a^2b^2c^2 + a'^2a^2(b^2+c^2-a^2) + b'^2b^2(c^2+a^2-b^2) + c'^2c^2(a^2+b^2-c^2) - a^2(a'^2-b'^2)(a'^2-c'^2) - b^2(b'^2-c'^2)(b'^2-a'^2) - c^2(c'^2-a'^2)(c'^2-b'^2)

  The six straight lines joining four points taken arbitrarily in a plane satisfy this relation.

• Polyhedrons

  p^2(2a^2b^2+2b^2c^2+2c^2a^2 -a^4-b^4-c^4) = -a^2b^2c^2 + a'^2a^2(b^2+c^2-a^2) + b'^2b^2(c^2+a^2-b^2) + c'^2c^2(a^2+b^2-c^2) - a^2(a'^2-b'^2)(a'^2-c'^2) - b^2(b'^2-c'^2)(b'^2-a'^2) - c^2(c'^2-a'^2)(c'^2-b'^2)

  With p the altitude of a tetrahedron on a base triangle of sides a, b, c and lateral edges a', b', c', the left side equals 144 V squared, giving the volume in terms of the six edges.

• Polyhedrons

  \cos{ADB}=\dfrac{\cos{\gamma}-\cos{\alpha'}\cos{\beta'}} {\sin{\alpha'}\sin{\beta}'}

  On the sphere, the cosine of the angle ADB equals (cos gamma minus cos alpha' cos beta') divided by sin alpha' sin beta'.

• Polyhedrons

  \cos{\alpha}=1-2\sin^2{\frac{\alpha}{2}}

  The cosine of an angle equals one minus twice the squared sine of half that angle.

• Polyhedrons

  aa'+bb'+cc'=2\sigma

  The quantity sigma is defined as half the sum of the products aa', bb', cc'.

• Polyhedrons

  36\hspace{3pt}V^2r^2=\sigma(\sigma-aa')(\sigma-bb')(\sigma-cc')

  The product 36 V squared r squared equals sigma times the three factors sigma minus aa', bb', cc', by analogy with Heron's formula.

Problems