PRISMS

568. A prism is a polyhedron two of whose opposite faces are equal polygons in parallel planes, and whose other faces are all parallelograms.

The bases of a prism are the equal, parallel polygons.

The lateral faces of a prism are the parallelograms.

The lateral edges of a prism are the intersections of the lateral faces.

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

The total area of a prism is the sum of the lateral area and the areas of the bases.

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

A triangular prism is a prism whose bases are triangles.

569. A right prism is a prism whose lateral edges are perpendicular to the planes of the bases.

A regular prism is a right prism whose bases are regular polygons.

An oblique prism is a prism whose lateral edges are not perpendicular to the planes of the bases.

A truncated prism is the portion of a prism included between the base and a plane not parallel to the base.

A right section of a prism is the section made by a plane perpendicular to the lateral edges of the prism.

570. A parallelepiped is a prism whose bases are parallelograms.

A right parallelepiped is a parallelepiped whose lateral edges are perpendicular to the planes of the bases.

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

An oblique parallelepiped is a parallelepiped whose lateral edges are not perpendicular to the planes of the bases.

A cube is a rectangular parallelepiped whose six faces are squares.

571. The unit of volume is a cube whose edges are each a unit of length.

The volume of a solid is the number of units of volume it contains. The volume of a solid is the ratio of that solid to

the unit of volume.

The three edges of a rectangular parallelepiped meeting at any vertex are the dimensions of the parallelepiped. Equivalent solids are solids that have equal volumes. Equal solids are solids that can be made to coincide.

Ex. What is the base of a rectangular parallelepiped? Of a right parallelepiped? Of an oblique parallelepiped?

NOTE. The space that is bounded by the surfaces of a solid, independ ent of the solid, is called a geometrical solid.

That is, if a material or physical body occupy a certain position and then be removed elsewhere, there is a definite portion of space that is the exact shape and size as the solid, and can be conceived as bounded by exactly the same surfaces as bounded the solid when in that original position. In order that we may pass planes and draw lines through solids, and superpose one solid upon another, it is convenient in studying the properties of solids to consider them usually as geometric solids, the material body being removed for the time.

The three kinds of parallelepipeds can be illustrated by removing the cover and bottom of an ordinary cardboard box, and distorting the shape of the frame that remains.

572. THEOREM.

573. THEOREM.

(See 511.)

PRELIMINARY THEOREMS

The lateral edges of a prism are all equal (?).

Any two lateral edges of a prism are parallel.

574. THEOREM. Any lateral edge of a right prism equals the altitude. (See 524.)

575. THEOREM. dicular to the bases.

576. THEOREM.

(Def. 569.)

577. THEOREM. piped are rectangles.

The lateral faces of a right prism are perpen(See 540.)

The lateral faces of a right prism are rectangles.

The faces and bases of a rectangular parallele(Def. 569.)

578. THEOREM. All the faces of any parallelepiped are parallelograms (?).

579. AXIOM. A polyhedron cannot have fewer than four faces.

580. AXIOM. A polyhedron cannot have fewer than three faces at each vertex.

THEOREMS AND DEMONSTRATIONS

581. THEOREM. The sections of a prism made by parallel planes

cutting all the lateral edges are equal polygons.

582. THEOREM. The opposite faces of a parallelepiped are equal

and parallel.

Ex. 2. Any section of a parallelepiped made by a plane cutting two

pairs of opposite faces is a parallelogram.

The lateral area = E· (AB +BC + CD + etc.) (Ax. 2). = E· perimeter of rt. sect. (Ax. 6).

Q.E.D.

Ex. 1. Any section of a parallelepiped made by a plane parallel to any edge is a parallelogram.

Ex. 2. The sum of the face angles at all the vertices of any parallelepiped is equal to 24 right angles.

Ex. 3. The sum of the plane angles of all the dihedral angles of any parallelepiped is equal to 12 right angles.

Proof: Pass three planes to three intersecting edges. Prove these sections whose are the plane angles of the dihedral angles, etc.

Ex. 4. Enunciate a theorem for the lateral area of a right prism.

Ex. 5. Find the lateral area of a right prism whose altitude is 8 feet

and each side of whose triangular base is 5 feet. 120 ft.

Ex. 6. Find the total area of a regular prism whose base is a regular hexagon, 10 inches on a side, if the altitude of the prism is 15 inches. 9007300 Ex. 7. Find the lateral area of a prism whose edge is 12 inches and whose right section is a pentagon, the sides of which are 3, 5, 6, 9, and 40% 11 inches.