A polyedron of four faces is called a tetraedron; one of six faces, a hexaedron; one of eight faces, an octaedron; one of twelve faces, a dodecaedron; one of twenty faces, an icosaedron. The following drawings show models of these polyedrons.

1. What is the least number of faces that a polyedron can have? Edges? Vertices? What kind of polyedron is it?

2. How many edges has a hexaedron? edron? An icosaedron?

3. How many vertices has a hexaedron? aedron? An icosaedron ?

An octaedron? A dodeca

An octaedron? A dodec

4. If E is the number of edges, F the number of faces, and V the number of vertices of a polyedron, show that in each of the five polyedrons named above E+2 F +V. = This principle is known as Euler's

Theorem.

5. Show that in a tetraedron, if S equals the sum of the face angles and V equals the number of vertices, S = 4(V - 2) right angles.

Is this formula true for a hexaedron? For an octaedron? dodecaedron? For an icosaedron ?

For a

349. Prisms. A prism is a polyedron of which two faces are congruent polygons in parallel

planes, and of which the other faces are parallelograms each of which has sides of the polygons as

two of its opposite sides.

The two congruent polygons in

RIGHT PRISM

parallel planes are called the bases, and the other faces are

called the lateral faces. The intersections of the lateral faces are called the lateral edges.

The perpendicular distance between the bases of a prism is called the altitude of the prism.

The sum of the areas of the lateral faces of a prism is called the lateral area.

The section of a prism which is made by a plane which intersects all of the lateral edges and is perpendicular to them is called a right section

edges are not perpendicular to its bases is called an oblique prism.

A prism is called triangular, quadrangular, hexagonal, etc., according as its bases are triangles, quadrilaterals, hexagons,

etc.

350. Fundamental properties of a prism. The following important properties of a prism are easily deduced from the definitions above. The student should draw figures and reason out the correctness of each.

(1) The lateral edges of a prism are parallel and equal.

(2) The altitude of a right prism is equal to a lateral edge of the prism.

(3) The lateral faces of a right prism are rectangles.

(4) Sections of a prism made by parallel planes cutting all lateral edges are congruent polygons.

(5) Right sections of a prism are congruent polygons.

EXERCISES

1. On a piece of cardboard, draw a figure similar to the adjoining figure, making each side of each triangle 3 in. and the rectangles 3 in. by 5 in. Cut out the pattern, and, by folding along the dotted lines and pasting, make a model of a triangular prism.

2. What kind of polyedrons are the cells which contain the honey in the comb of the bee? What is the advantage of building the cells in this form?

3. In the form of what kind of polyedron are lead pencils sometimes made?

4. A rectangular room or box is what kind of prism?

5. Glass prisms, such as that shown in the drawing, are used in optical instruments for changing the direction of light passing through them. A beam of white light passing through this prism is dispersed or separated into a rainbow-colored band, gradually changing from red at one end, through orange,

vibgyor

yellow, etc., to violet at the other end. What kind of prism is it? Prismatic pieces of glass are used also as pendants or fringes on chandeliers to disperse the light.

6. The drawing shows a model of a polyedron of which two faces are congruent polygons in parallel planes, and of which the other faces are all parallelograms. Why is it not a prism?

7. A section of a prism made by a plane parallel to a base is congruent to the base.

8. Any section of a prism made by a plane parallel to a lateral edge is a parallelogram.

9. The lateral faces of a right prism are perpendicular to the bases.

A right parallelopiped whose bases are rectangles is a rectangular parallelopiped.

A rectangular parallelopiped all of whose faces are squares is a cube.

353. Fundamental properties of a parallelopiped. — The following properties of a parallelopiped follow from the definitions above. The student should draw figures and reason. out the correctness of each.

(1) The opposite lateral faces of a parallelopiped are congruent and parallel.

(2) Any two opposite faces of a parallelopiped may be taken as the bases.

(3) Any right section of a parallelopiped is a parallelogram.

EXERCISES

1. Find the sum of all of the face angles of any parallelopiped.

2. The diagonals of a rectangular parallelopiped are equal.

3. How, by measuring the diagonals of a piece of house framing, can a carpenter tell when it is truly rectangular?

4. The square of a diagonal of a rectangular parallelopiped is equal to the sum of the squares of three concurrent edges.

5. Find the diagonal of a cube whose edge is 2 in.

6. If the edge of a cube is e, find the length of a diagonal of the cube.

7. The diagonal of a face of a cube is 3√2. Find the diagonal of the cube.

8. Are the diagonals of a cube perpendicular to each other?

9. A suitcase is 26 in. long, 15 in. high, and 7 in. thick. Can an umbrella which is 32 in. long be packed inside of it?

10. Show that the edge, diagonal of a face, and diagonal of a cube are in the ratio of 1: √2: √3.

11. The sum of the squares of the four diagonals of any parallelopiped is equal to the sum of the squares of the twelve edges.

12. The diagonals of any parallelopiped all meet at one point, which is the middle point of each.

13. The intersection of the diagonals of any parallelopiped and the intersections of the diagonals of two opposite faces are in a straight line.

354. Theorem. The lateral area of a prism is equal to the product of a lateral edge and the perimeter of a right section.

Hypothesis. MNOP... is a right section of a prism, and AF is a lateral edge.

Conclusion. The lateral area = AF(MN+NO+OP+etc.).

Suggestions. Express the

area of each lateral face, and

form the sum of these areas.

Apply § 350 (1). Factor.