Therefore As ABC and A'B'C' are identically equal, and can be superposed.

(Art. 53.)

Suppose now that the section P'A'B'C' is placed upon the section PABC so that the points A', B', C' coincide with their corresponding points A, B, C, and let A, A' and C, C' remain. fixed while B traverses the section PABC. Since B will always coincide with its corresponding point B', every point of the one section will coincide with its corresponding point in the other.

Therefore the two sections are identically equal.

553. COROLLARY I. The bases of a cylinder are identically equal.

554. COROLLARY II. Any two right sections of a cylinder are identically equal.

555. COROLLARY III. Any section of a cylinder parallel to the base is identically equal to the base.

556. COROLLARY IV. All sections of a circular cylinder parallel to its bases are equal circles, and the straight line joining the centres of the bases passes through the centres of all the parallel sections.

EXERCISES

1. If through any point of the lateral surface of a cylinder a straight line is drawn parallel to any element of the surface, this straight line is also an element of the surface.

2. Show that all the elements of the lateral surface of a cylinder are of equal lengths.

3. The line of intersection of two planes tangent to a cylinder is parallel to an element of the lateral surface, and to the plane through their two elements of contact.

4. Find the locus of points (1) at a given distance from a given straight line; (2) at given distances from each of two given parallel straight lines.

DEFINITIONS

557. A prism is inscribed in a cylinder when its bases and the bases of the cylinder lie in the same planes, and its lateral edges are elements of the lateral surface of the cylinder.

The cylinder is at the same time circumscribed about the prism.

The section of an inscribed prism made by any plane is inscribed in the section of the cylinder made by the same plane.

558. A prism is circumscribed about a cylinder when its bases and the bases of the cylinder lie in the same planes, and its lateral faces are tangent to the cylinder.

The cylinder is at the same time inscribed in the prism.

The section of a circumscribed prism made by any plane is circumscribed about the section of the cylinder made by the same plane.

Prism inscribed in a cylinder

559. If a regular prism is inscribed, or circumscribed, to a circular cylinder, and the number of its lateral faces is indefinitely increased in some regular way, the lateral surface of the prism approaches the lateral surface of the cylinder as its limit, the volume of the prism approaches the volume of the cylinder as its limit, and the perimeter and area of a right section of the prism approach the perimeter and area of a right section of the cylinder as their limits.

PROPOSITION II

560. The lateral area of a circular cylinder is equal to the product of the perimeter of a right section of the cylinder and the length of an element of its lateral surface.

Let PQ be a circular cylinder of which S is the perimeter of a right section, and AB an element of the lateral surface.

It is required to prove that the lateral area of PQ is equal to S times AB.

Proof. In the cylinder PQ inscribe a regular prism AH, one of whose elements is AB. Let KM be a right section of this prism. The lateral area of AH = perimeter of KM × AB. (Art. 479.)

If the number of lateral faces of AH is indefinitely increased, the lateral area of AH approaches the lateral area of PQ as its limit, and the perimeter of KM approaches S as its limit. Therefore the lateral area of PQ = S × AB. (Art. 230.)

561. COROLLARY. The lateral area of a right circular cylinder is equal to the product of the altitude and the circumference of the base.

If the lateral area is represented by A, the radius of the base by r, and the altitude by h,

A = 2 Trh.

PROPOSITION III

562. The volume of a circular cylinder is equal to the product of its altitude and the area of its base.

The proof of this theorem is similar to that of Proposition II, and is left to the pupil. Reference should be made to

Article 507.

If the volume is represented by V, the altitude by h, and the radius of the base by r,

563. COROLLARY.

V = πr2h.

The volumes of all circular cylinders, whether right or oblique, having equal bases and equal altitudes are equal.

EXERCISES

NOTE. Use 34 as the approximate value of π.

1. Find the volume, the lateral area, and the total area of a right circular cylinder the diameter of whose base is 14 inches, and whose altitude is 11 inches.

2. The lateral area of a right circular cylinder is 528 square feet, and its volume is 1584 cubic feet. Find the diameter and circumference of its base, and its altitude.

3. Two right circular cylinders are of equal height while the circumference of one is double the circumference of the other. What is the ratio of their lateral areas? Also of their volumes?

4. A hollow cylindrical iron tube has an outer diameter of 12 inches and an inner diameter of 9 inches. How many cubic inches of iron are there in a piece of the tube 2 feet long?

5. When a tap is opened the water in a pipe of one inch inner diameter flows at the rate of two miles an hour. How many gallons of water would flow from the tap in 20 minutes?

Take 7 gallons to a cubic foot.

6. Show that the volumes of two similar right circular cylinders are in the same ratio as the cubes of their altitudes, or as the cubes of the radii of their bases.

7. What is the altitude of a right circular cylinder if its lateral area equals the sum of the areas of its bases?

SECTION II

THE CONE

564. DEFINITION. If a straight line moves so as always to pass through a fixed point, while some point of the line traverses a fixed curve not in a plane with the fixed point, it will describe a conical surface.

As in the cylindrical surface, the generating line in any position is an element of the surface. The directing curve will always be considered a closed curve.

The fixed point through which the generating line always passes is called the vertex of the surface.

Conical surface

If the generating line is indefinite in length, the surface is divided into two parts at the vertex, called the two sheets, or the two nappes of the conical surface. When we speak of a plane section of a conical surface, we shall always have in mind the section made by a plane which cuts all the elements on the same side of the vertex, that is, a section of one of the sheets.

565. DEFINITION. A cone is a figure consisting of a plane surface and a conical surface intercepted

between the plane surface and the vertex.

The plane surface is called the base, and the conical surface the lateral surface. The vertex of the conical surface is the vertex of the cone.

The straight line joining any point of the lateral surface to the vertex is an element of the lateral surface, since it must coincide with the generating line in one position.

A circular cone is one whose base is circular.

Circular cone

The straight

line joining the vertex to the centre of the base is called the

axis of the cone.