PROPOSITION XXV. THEOREM

986. Similar tetrahedrons are to each other as the cubes of their homologous edges.

A

B

B'

Let O-ABC and o'-A'B'C' be two similar tetrahedrons.

987. EXERCISE. The homologous edges of two similar tetrahedrons are 3 in. and 4 in. respectively. The volume of the former is 50 cu. in. Find the volume of the other.

988. EXERCISE. The volume of a given tetrahedron is 40 cu. ft. Construct a similar tetrahedron whose volume shall be 5 cu. ft.

989. EXERCISE. From a given tetrahedron cut off a frustum whose volume shall equal 2§ of the given tetrahedron.

PROPOSITION XXVI. THEOREM

990. Similar polyhedrons are to each other as the cubes of their homologous edges.

Let AD-L and A'D'-L' be two similar polyhedrons.

Proof. Divide the polyhedrons into similar tetrahedrons having the common vertices M and M'.

Designate the tetrahedrons of AD-L by T1, T2, T3, T1, etc., and the tetrahedrons of A'D'-L' by T'1, T'2, T'з, T', etc.

991. EXERCISE. The volume of a certain polyhedron is 135 cu. yds. Construct a polyhedron similar to the given polyhedron, and having a volume 40 cu. yds. Compare the surface of the constructed polyhedron with that of the given one.

992. DEFINITION. A regular polyhedron is a polyhedron whose faces are equal regular polygons, and whose polyhedral angles are equal.

993. THE NUMBER OF REGULAR POLYHEDRONS. Each polyhedral angle has three or more faces, and the sum of its plane angles is less than 4 R.A.'s.

Show that the number of equilateral triangles that can be used to form a polyhedral angle is three, four, or five.

Show that the number of squares that can be used to form a polyhedral angle is three.

Show that the number of regular pentagons that can be used to form a polyhedral angle is three.

Show that regular polygons having more than five sides cannot be used to form a polyhedral angle.

There are, therefore, five regular polyhedrons.

Three of

these are bounded by equilateral triangles, one by squares, and one by regular pentagons.

Regular Tetrahedron

Regular Hexahedron

Regular Octahedron

The regular tetrahedron is bounded by four equilateral triangles.

The regular hexahedron (or cube) is bounded by six squares. The regular octahedron is bounded by eight equilateral triangles.

The regular dodecahedron is bounded by twelve regular pentagons.

The regular icosahedron is bounded by twenty equilateral triangles.

To construct the regular polyhedrons cut out cardboard as indicated in the following diagrams. Fold on the broken lines and bring the edges together.

A

XXX

EXERCISES

1. The lateral surface of a pyramid is greater than its base.

2. In any rectangular parallelopiped the square of a diagonal is equal to the sum of the squares of three edges that meet at a common vertex.

3. The altitude of a pyramid is divided into four equal parts by planes parallel to the base. Find the ratio to one another of the four solids into which the pyramid is divided.

4. The volume of a right prism is 480 cu. ft. Its base is a R. A. triangle whose legs are 16 ft. and 12 ft. Find its lateral area.

5. The diagonal of a cube is equal to the product of its edge by √3. 6. The volume of a cube is 219 cu. in. Find the length of its diagonal.

7. In a tetrahedron planes passed through the three lateral edges and the middle points of the sides of the base pass through a common line.

8. The volume of a regular tetrahedron is equal to the cube of an edge multiplied by√2.

9. Find the surface and volume of a regular tetrahedron whose edge is 4 in.

10. The diagonals of a rectangular parallelopiped are equal, and pass through a common point.

11. The lines joining the points of intersection of the diagonals of the opposite faces of a rectangular parallelopiped pass through a common point.

12. If E, F, G, and H are the middle points of the edges AB, AD, CD, and BC respectively of the tetrahedron ABCD, prove EFGH a parallelogram.

13. The volume of a regular prism is equal to the product of its lateral area by one half the apothem of its base.

14. The areas of the bases of the frustum of a pyramid are 15 sq. in. and 50 sq. in. The altitude of the frustum is 7 in. Find the altitude of the pyramid.

15. The base of a pyramid is a square, and its lateral faces are equilateral triangles. If its altitude is 6 ft., find the volume and lateral area.

16. The lines joining each vertex of a tetrahedron with the point of intersection of the medial lines of the opposite face all meet in a common point, which divides each line in the ratio 1:4. [The point of intersection is the center of gravity of the tetrahedron.]

17. In any parallelopiped the sum of the squares of the four diagonals is equal to the sum of the squares of the twelve edges.

18. In a rectangular parallelopiped three of the edges are 8 in., 9 in., and 12 in. respectively. Find the length of a diagonal of the parallelopiped.

19. The diagonal of a cube is a inches. Find its volume.

20. Any line through the point of intersection of the diagonals of a parallelopiped, and terminating in the surface, is bisected at that point.