cae is half one of the n equal angles formed on the sphere round a,

137. To find the radii of the inscribed and circumscribed· spheres of a regular polyhedron.

Let the edge AB=a, let OC=r and OA = R, so that r is the radius of the inscribed sphere, and R the radius of the circumscribed sphere. Then

138. To find the surface and volume of a regular polyhedron.

Also the volume of the pyramid which has one face of the

polyhedron for base and O for vertex is

therefore the volume of the polyhedron is

r ma2 π cot and 3 4

m

139. To find the volume of a parallelopiped in terms of its edges and their inclinations to one another.

B

=

Let the edges be OA = a, OB=b, OC=c; let the inclinations be BOC a, COA=B, AOB = y. Draw CE perpendicular to the plane AOB meeting it in E. Describe a sphere with O as a centre, meeting OA, OB, OC, OE in a, b, c, e respectively.

The volume of the parallelopiped is equal to the product of its base and altitude ab sin y. CE abc sin y sin cOe. The spherical triangle cae is right-angled at e; thus

=

=

sin coe = sin coa sin cae =

and from the spherical triangle cab

sin ẞ sin cab,

√(1 - cos3 a - cos2 ß - cos3 y + 2 cos a cos ß cos

sin ẞ sin y

therefore the volume of the parallelopiped

= abc √(1 - cos3 a - cos2 ß - cos3 y + 2 cos a cos ẞ cos y).

140. To find the diagonal of a parallelopiped in terms of the three edges which it meets and their inclinations to one another.

Let the edges be OA = a, OB = b, OC =c; let the inclinations be BOC=a, COA =B, AOB = y. Let OD be the diagonal required, and let OE be the diagonal of the face OAB.

OD2 = OE2 + ED2 + 20E. ED cos COE

=

Then

a2 + b2+2ab cos y + c2 + 2cOE cos COE.

Describe a sphere with O as centre meeting OA, OB, OC, OE in a, b, c, e respectively; then (see example 14, Chap. IV.)

therefore OD2 = a2 + b2 + c2 + 2ab cos y + 2bc cos a + 2ca cos ß.

141. To find the volume of a tetrahedron.

A tetrahedron is one-sixth of a parallelopiped which has the same altitude and its base double that of the tetrahedron; thus if

the edges and their inclinations are given we can take one-sixth of the expression for the volume in Art. 139. The volume of a tetrahedron may also be expressed in terms of its six edges; for in the figure of Art. 139 let BC = a', CA=b'′, AB = c; then.

and if these values are substituted for cos a, cos ẞ, and cos sy in the expression obtained in Art. 139, the volume of the tetrahedron will be expressed in terms of its six edges.

EXAMPLES.

1. If I denote the inclination of two adjacent faces of a regular polyhedron, shew that cos I = in the tetrahedron, =0 in the cube, in the octahedron, =- √5 in the dodecahedron, and -5 in the icosahedron.

2. With the notation of Art. 136, shew that the radius of the sphere which touches one face of a regular polyhedron and all

the adjacent faces produced is a cot #cot I.

m

3. A sphere touches one face of a regular tetrahedron and the other three faces produced; find its radius.

4. If a and b are the radii of the spheres inscribed in and described about a regular tetrahedron, shew that b = 3a.

5. If a is the radius of a sphere inscribed in a regular tetrahedron, and R the radius of the sphere which touches the edges, shew that R2 3a2.

=

6. If a is the radius of a sphere inscribed in a regular tetrahedron, and R the radius of the sphere which touches one face and the others produced, shew that R = 2a.

7. If a cube and an octahedron be described about a given sphere, the sphere described about these polyhedrons will be the same; and conversely.

8. If a dodecahedron and an icosahedron be described about a given sphere, the sphere described about these polyhedrons will be the same; and conversely.

9. A regular tetrahedron and a regular octahedron are inscribed in the same sphere; compare the radii of the spheres which can be inscribed in the two solids.

10. The sum of the squares of the four diagonals of a parallelopiped is equal to four times the sum of the squares of the edges.

11. If with each angular point of any parallelopiped as centres equal spheres be described, the sum of the intercepted portions of the parallelopiped will be equal in volume to one of the spheres.

12. A regular octahedron is inscribed in a cube so that the corners of the octahedron are in the centres of the faces of the cube; prove that the volume of the cube is six times that of the octahedron.

13. It is not possible to fill any given space with a number of regular polyhedrons of the same kind, except cubes; but this may be done by means of tetrahedrons and octahedrons which have equal faces, by using twice as many of the former as of the latter.

14. A spherical triangle is formed on the surface of a sphere of radius p; its angular points are joined, forming thus a pyramid with the lines joining them with the centres; shew that the volume of the pyramid is

3

where r, r,,,, r, are the radii of the inscribed and escribed circles of the triangle.

15. The angular points of a regular tetrahedron inscribed in a sphere of radius r being taken as poles, four equal small circles of the sphere are described, so that each circle touches the other