REGULAR POLYEDRONS

688 DEFINITION. A regular polyedron is a polyedron whose faces are equal regular polygons, and whose polyedral angles are equal.

PROPOSITION XXVII. THEOREM

689 Only five regular convex polyedrons are possible.

PROOF. A convex polyedral angle must have not less than three faces, and the sum of its face angles must be less than 360°. § 591

1 Each angle of an equilateral triangle is 60°. The sum of three, four, or five such angles is less than 360°, but the sum of six such angles is 360°. Hence three, four, or five equilateral triangles may form a convex polyedral angle, but six or more such triangles cannot form a solid angle.

§ 591 Therefore only three regular convex polyedrons are possible whose surfaces are composed of triangles.

2 Each angle of a square is 90°. The sum of three such angles is less than 360°, but the sum of four such angles is 360°.

Therefore only one regular convex polyedron is possible whose surface is composed of squares.

3 Each angle of a regular pentagon is 108°. The sum of three such angles is less than 360°, but the sum of four such angles is greater than 360°.

Therefore only one regular convex polyedron is possible whose surface is composed of pentagons.

4 Each angle of a regular hexagon is 120°. The sum of three such angles is 360°. Hence a convex polyedral angle cannot be formed by regular polygons of six or more sides.

Therefore only five regular convex polyedrons are possible.

Q. E. D.

690 The five regular polyedrons are the tetraedron, the hexaedron, the octaedron, the dodecaedron, and the icosaedron.

They may be constructed as follows: Draw the following figures on thin cardboard. Cut out the figures, and cut half through the cardboard on the dotted lines. Fold along the dotted lines, and fasten the edges together by pasting strips of paper along them.

A + ~

1

X

Tetraedron Hexaedron Octaedron Dodecaedron Icosaedron

CYLINDERS

DEFINITIONS

691 A cylindrical surface is a curved surface generated by a moving straight line which always remains parallel to its original position and which constantly touches a fixed curved line. The generatrix is the moving straight line.

The directrix is the fixed curved line. An element of the cylindrical surface is the generatrix in any position.

NOTE. The generatrix is usually considered indefinite in extent. The directrix may be any curve whatever, but closed curves, usually circles, are the only ones considered in Elementary Geometry.

692 A cylinder is a solid bounded by a cylindrical surface and two parallel plane surfaces.

The lateral surface of a cylinder is its cylindrical surface. The bases of a cylinder are its two plane surfaces.

693 COROLLARY. All elements of a cylinder are equal.

694 The altitude of a cylinder is the perpendicular between the planes of its bases.

695 A section of a cylinder is the figure formed by a plane intersecting the cylinder. A right section of a cylinder is a section perpendicular to the elements.

696 A right cylinder is a cylinder whose elements are perpendicular to its bases. An oblique cylinder is a cylinder whose elements are oblique to its bases.

697 A circular cylinder is a cylinder whose bases are circles.

698 A cylinder of revolution is a cylinder generated by the revolution of a rectangle about one side as

an axis.

Such a cylinder is a right circular cylinder.

699 Similar cylinders of revolution are cylinders generated by the revolution of similar rectangles about their homologous sides as axes.

700 A tangent line to a cylinder is an indefinite straight line. which touches the lateral surface in one point only.

701 A tangent plane to a cylinder is an indefinite plane which contains an element of the cylinder without cutting its surface. The element contained by a tangent plane is called the element of contact.

702 A prism is inscribed in a cylinder when its lateral edges are elements of the cylinder and its bases are inscribed in the bases of the cylinder.

703 A prism is circumscribed about a cylinder when its lateral faces are tangent to the cylinder and its bases are circumscribed about the bases of the cylinder.

PROPOSITION XXVIII. THEOREM

704 Every section of a cylinder made by a plane passing through an element is a parallelogram.

B

A

D

HYPOTHESIS. ABCD is a section of the cylinder BD made by a plane passing through the element AB.

CONCLUSION. ABCD is a parallelogram.

PROOF

Let DC' be the element through D.

Then DC' is to AB.

.. DC' lies in the plane AC.

§ 691

§§ 515, 517

Since DC' is common to the plane AC and the cylindrical surface, it must be their intersection and coincide with DC.

.. DC is parallel to AB. Also, BC is parallel to AD.

§ 543

.. ABCD is a parallelogram.

§ 195

Q. E. D.

705 COROLLARY. Every section of a right cylinder made

by a plane passing through an element is a rectangle.