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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.

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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.

561. COROLLARY.

(Art. 230.)

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 = r2h.

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.

When the vertex lies on the perpendicular to the base drawn from its centre, the cone is called a right circular cone.

566. The altitude of a cone is the perpendicular distance from the vertex to the plane of the base.

567. The slant height of a right circular cone is the length of an element of the lateral surface.

568. THEOREM I. Any plane which contains one element of the lateral surface of a cone and meets the surface at any other point, contains also a second element, and the section of the cone by the plane is a triangle.

If the plane L contains the element SB of the cone S-ABC, and meets the surface again at the point D it must contain the element SD, since it contains two points of this element.

The section of the cone by the plane L therefore consists of the line-segments SB, SD, and BD, and is thus a triangle.

569. COROLLARY.

The section of a right circular cone by a plane through the vertex is an isosceles triangle.

B

L

570. DEFINITION. If a plane contains one element of the lateral surface of a cone and only one, it is said to be tangent to the cone, and the element which it contains is called the element of contact.

571. THEOREM II. The plane determined by a tangent to the base of a circular cone and the element of the lateral surface passing through its point of contact is tangent to the cone.

CONVERSELY. If a plane is tangent to a circular cone, its intersection with the plane of the base is tangent to the base.

See Article 550.

PROPOSITION IV

572. Any section of a circular cone made by a plane parallel to the base is a circle, and its centre lies upon the straight line joining the vertex to the centre of the base.

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Let A'B'Q' be the section of the circular cone S-ABQ made by a plane L parallel to the base, and let SO be the straight line joining S to the centre of the base.

It is required to prove that A'B'Q' is a circle whose centre lies on SO.

Proof. Let A' and B' be any two points of the section, and let the planes determined by the line SO and A', SO and B' intersect the lateral surface of the cone in the straight lines SA'A, SB'B, the base in the lines AO, BO, and the plane L in the lines A'O', B'O', parallel to 40 and BO, respectively.

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Since A' and B' are any points whatever in the section made by L, all points of the section are equidistant from O'.

Therefore the section is a circle with its centre at O'.

573. COROLLARY. The radii of two sections of a circular cone, parallel to the base, are in the same ratio as their distances from the vertex.

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