69. A section of a pyramid made by a plane parallel to the base is equal to one ninth of the base in area, and the altitude of the pyramid is 6 ft. How far from the vertex is the plane of the section?

70. At what distance from the vertex must a pyramid whose altitude is 12 in. be cut by a plane parallel to the base in order to divide it into two equivalent parts?

71. The length of a lateral edge of a pyramid is a. At what distances from the vertex will this edge be cut by two planes parallel to the base, which divide the pyramid into three equivalent parts?

72. A regular triangular pyramid has a for its altitude and for each side of its base. Find the area of a section parallel to the base and distance a from the vertex; also find the volume of the pyramid.

73. The stone cap of a gate post is in the form of a regular quadrangular pyramid whose base measures 4 in. on a side and whose altitude is 15 in. If the top of the cap is cut off by a plane parallel to the base and 5 in. above it, what is the volume of the piece cut off?

74. The altitude of a frustum of a pyramid is 16 in., and two homologous sides of its bases are 12 in. and 10 in. Find the ratio of the volume of the frustum to the volume of the entire pyramid.

75. Two similar pyramids have altitudes of 6 ft. and 8 ft. Find the ratio of their surfaces, and also of their volumes.

76. The height of the Great Pyramid is 489 ft. An exact model of the pyramid is made of height 4.89 ft., its side faces being triangles similar to the side faces of the pyramid. What is the ratio of the total lateral area of the pyramid to that of the model?

77. The volumes of two similar polyhedrons are respectively 3 cu. in. and 24 cu. in., and one edge of the first is 5 in. What is the homologous edge of the second?

78. The volumes of two similar polyhedrons are respectively 64 cu. ft. and 216 cu. ft. If the total area of the first polyhedron is 112 sq. ft., what is the total area of the second?

79.

The base of a right pyramid is a regular hexagon. The altitude is h and the length of an edge of the base is a. Find the area of the

base of a similar pyramid having one eighth of the volume.

80. The altitude of a certain solid is 2 in., its lateral area is 15 sq. in., and its volume is 4 cu. in.

Find the altitude and lateral area of

a similar solid whose volume is 256 cu. in.

BOOK VIII

THE CYLINDER AND THE CONE

THE CYLINDER

A cylindrical surface is a curved surface generated by a straight line which continually moves along a fixed curve and is constantly parallel to a given straight line not coplanar

with the curve.

For example, if the straight line CF moves along the curve CDE so that in every posi

tion it is parallel to AB, the surface thus generated is a cylindrical surface.

The moving line is called the generatrix, and the curve is called the directrix. The generatrix in any position is called an element of the surface.

E

H

G

B

F

A cylinder is a solid bounded by a closed cylindrical surface and two parallel planes. The sections of the cylindrical surface formed by the parallel planes are called the bases of the cylinder, and the cylindrical surface is called the lateral surface. The area of the lateral surface is called the lateral area of cylinder. An element of the cylindrical surface is called an element of the cylinder.

A cylinder whose elements are perpendicular to its bases is called a right cylinder. A cylinder whose elements are oblique to its bases is called an oblique cylinder.

A cylinder whose bases are circles is called a circular cylinder. The line joining the centres of the bases is called the axis of the circular cylinder.

The perpendicular distance between the bases is called the altitude of the cylinder.

A section of a cylinder made by a plane perpendicular to the elements is called a right section.

A plane is tangent to a cylinder when it passes through an element, but does not meet it again, however far it is produced. The element through which the plane passes is called the element of contact.

The following principles follow at once from the preceding definitions:

(i)

The elements of a cylinder are parallel and equal. (ii) Any element of a right cylinder is equal to the altitude.

(iii) A line drawn through any point in a cylindrical surface parallel to an element is an element.

Proposition 282 Theorem

by

A right circular cylinder may be generated the revolution of a rectangle about one of its sides as an axis.

Prove that the solid generated fulfils all the requisites of a right circular cylinder. Consult Prop. 210 and Prop. 225.

DEFINITIONS. On account of the manner in which a right circular cylinder may be generated, it is also called a cylinder of revolution. The radius of the base is called the radius of the cylinder.

Similar cylinders of revolution are cylinders generated by the revolution of similar rectangles about homologous sides

as axes.

A cylinder of revolution whose section through the axis is a square is called an equilateral cylinder.

Ex. 1137. The axis of a circular cylinder is equal and parallel to all the elements.

Proposition 283 Theorem

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

Hypothesis. ABCD is a section of the cylinder HK, made by a plane passing through the element AD.

Conclusion. ABCD is a .

Proof. Through B draw BE in the plane AC || to AD. Then BE is an element of the cylinder.

(?)

.. BE is the intersection of the plane and the cylindrical

COR. Every section of a right cylinder made by a plane passing through an element is a rectangle.

Proposition 284 Theorem

The bases of a cylinder are congruent.

Hypothesis. ABC and DEF are the bases of the cylinder HK.

Conclusion. Base ABC = base, DEF.

Proof. Let A, B, and C be any three points in the perimeter of base ABC, and let the elements passing through A, B, and C be AD, BE, and CF respectively. Draw AB, AC, BC, DE, DF, and EF.

Apply base ABC to base DEF so that ▲ ABC shall coincide with A DEF. Then A, B, and C will fall on D, E, and F respectively.

Since A, B, and C are any three points in the perimeter of base ABC,every point in the perimeter of base ABC will fall on the perimeter of base DEF.

.. base ABC = base DEF.

(?)

Q. E.D.

COR. I. The sections of a cylinder made by parallel planes which cut all the elements are congruent.