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.

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

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 I 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.

PROPOSITION XXIX.

THEOREM

706 The bases of a cylinder are equal..

E

0000

HYPOTHESIS. ABC and DEF are the bases of the cylinder AF.
CONCLUSION. ABC= DEF.

PROOF

Through any element AD pass any two planes forming the sections AE and AF. Join BC and EF.

Then AE and AF are S

§ 704

.. BE and CF are equal and parallel.

§ 536

.. BF is a, and BC= EF, AB=DE, AC= DF.

§ 200

.. Δ ABC = Δ DEF.

$172

Superpose the bases, making the equal ▲ coincide.

Since A, B, C, are any points in the perimeter of the lower base, every point in the perimeter of the lower base will fall in the perimeter of the upper base.

..the bases will coincide and are equal.

Q. E. D.

707 COROLLARY 1. Any two parallel sections cutting all the elements of a cylinder are equal.

For these sections are the bases of a cylinder.

708 COROLLARY 2. Any section of a cylinder parallel to the base is equal to the base.

709 If a prism whose base is a regular polygon is inscribed in or circumscribed about a circular cylinder, and if the number of sides of the base of the prism is indefinitely increased,

The lateral area of the prism approaches the lateral area of the cylinder as a limit.

The volume of the prism approaches the volume of the cylinder as a limit.

A right section of the prism approaches a right section of the cylinder as a limit.

PROOF

If the number of lateral faces of either prism is indefinitely increased, its bases will approach the bases of the cylinder as their limits. Therefore,

§ 455 The lateral area of the prism approaches the lateral area of the cylinder as a limit.

The volume of the prism approaches the volume of the cylinder as a limit.

A right section of the prism approaches a right section of the cylinder as a limit.

Q. E. D.