574. DEFINITIONS.

A pyramid is inscribed in a cone when

their bases lie in the same plane, and the lateral edges of the pyramid are elements of the lateral surface of the cone.

The vertex of the inscribed pyramid coincides with the vertex of the cone, and the base of the pyramid is inscribed in the base of the cone. The section of the inscribed pyramid made by any plane is inscribed in the section of the cone made by the same plane.

575. A pyramid is circumscribed about a cone when their bases lie in the same plane, and the lateral faces of the pyramid are tangent to the cone.

The vertex of the circumscribed pyramid coincides with the vertex of the cone, and the base of the pyramid is circumscribed about the base of the cone. The section of the circumscribed pyramid made by any plane is circumscribed about the section of the cone made by the same plane.

576. If a regular pyramid is inscribed, or circumscribed, to a circular cone and the number of its lateral faces is indefinitely increased in some regular way, the lateral surface of the pyramid approaches the lateral surface of the cone as its limit, and the volume of the pyramid approaches the volume of the cone as its limit.

PROPOSITION V

577. The lateral area of a right circular cone is equal to one-half the product of the circumference of the base and the slant height.

S

E

Let S-ACE be a right circular cone, of which is the slant height, and M the circumference of the base.

It is required to prove that the lateral area of S-ACE is equal to Ml.

Proof. Inscribe in the cone S-ACE a regular pyramid P, one of whose elements is SA = 1, and the perimeter of whose base is H.

=

Hl.

(Art. 512.)

Then the lateral area of P If the number of lateral faces of P is indefinitely increased, the lateral area of P approaches the lateral area of S-ACE as its limit, and the perimeter H approaches the circumference M as its limit.

Therefore the lateral area of S-ACE

=

MI.

(Art. 230.)

If the radius of the base is r, and the lateral area is represented

by A,

A = πrl.

578. DEFINITION. The frustum of a cone is that portion of a cone intercepted between the base and a plane parallel to the base intersecting the lateral surface.

PROPOSITION VI

579. The volume of a circular cone is equal to onethird the product of its altitude and the area of its base.

The proof of this theorem is similar to that of Proposition V, and is left to the pupil.

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

V = }πr2h.

MISCELLANEOUS EXERCISES

1. Find the lateral area and the volume of a right circular cone, the area of whose base is 154 square inches and whose altitude is 11 inches.

2. The slant height of a right circular cone is 4 metres. How far from the vertex must a section parallel to the base be taken so as to divide the lateral area into two equal parts?

3. A right triangle whose sides are 3, 4, 5 feet, respectively, is rotated about the shortest side. Find the area of the surface described by the hypotenuse.

4. Show that the lateral areas of similar right circular cylinders are in the same ratio as the squares of their altitudes, or as the squares of the radii of their bases.

DEFINITION. Similar right circular cylinders are generated by the rotation of similar rectangles about homologous sides:

5. Show that the lateral areas of two similar cones of revolution are in the same ratio as the squares of their slant heights, or as the squares of their altitudes, or as the squares of the radii of their bases.

DEFINITION. Similar cones of revolution, or right circular cones, are generated by similar right triangles rotating about homologous sides.

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

7. The volumes of two similar cones of revolution are in the ratio of 512.729. What is the ratio of their lateral areas?

8. Show that the lateral area of the frustum of a right circular cone is equal to the sum of the circumferences of its bases multiplied by one-half the slant height.

9. Show that the lateral area of the frustum of a right circular cone is equal to the circumference of a section midway between the bases multiplied by the slant height.

10. Show that the volume of the frustum of a circular cone, the areas of whose bases are B and b and whose altitude is h, is given by the formula

See Ex. 2, p. 338.

V = {h (B + b + √Bb).

11. A right circular cylinder of height 2 ft. and the radius of whose base is 6 in. rolls on a plane making one complete revolution. What is the shape of the surface covered by it? Find its area.

12. A right circular cone whose altitude is 12 in. and the radius of whose base is 5 in. rolls on a plane, without slipping, making one complete revolution. What is the shape of the surface covered? Find its area.

13. A regular hexagonal prism is inscribed in a right circular cylinder. Compare their lateral areas and their volumes.

14. The base of a right circular cylinder has a radius of 7 cm. and an altitude of 7 cm. Find its total surface area and its volume.

15. A rectangle whose adjacent sides are 5 ft. and 7 ft., respectively, revolves in succession about these sides; show that the volumes of the cylinders generated are in the ratio 7: 5.

16. The total surface of a right circular cone is 462 sq. cm., and the slant height is twice the radius of the base; find the volume.

17. In a right circular cylinder of height 3 ft. and 8 in. radius, a square prism is inscribed. Find its volume.

18. A square whose side is 50 cm. revolves about one of its diagonals; find the area and volume of the figure so generated.

19. Show that in a right circular cone all the elements of the lateral surface are equal in length.

20. Show that the total area T of a right circular cylinder, including the areas of the two bases, is given by the formula :

T=2πrh + 2 πr2 = 2 πr (h + r).

21. A circular cistern is 22 ft. in circumference at the top, 16 ft. at the bottom, and 8 ft. deep. How many gallons of water will it hold, assuming 7 gallons to the cubic foot?

1. DEFINITIONS.

SUMMARY OF CHAPTER VIII

(1) Cylindrical Surface· the surface described by a straight line moving parallel to a fixed straight line, while some point of it traverses a fixed curve not in a plane with the fixed line. § 543. (2) Conical Surface the surface described by a straight line which moves so as always to pass through a fixed point, while some point of it traverses a fixed curve not in a plane with the fixed point. § 564.

(3) Closed Cylindrical Surface ·

is closed. § 543.

(4) Vertex of a Conical Surface ·

one for which the directing curve

the point through which all the

elements of the surface pass. § 564.

(5) Cylinder -a figure consisting of two parallel plane surfaces and a closed cylindrical surface intercepted between them. § 544. (6) Circular Cylinder - one whose bases are circular. § 544. (7) Right Cylinder - one in which the elements of the lateral surface

are perpendicular to the bases. § 544.

(8) Cone-a figure consisting of a plane and the conical surface intercepted between it and the vertex. § 565.

(9) Circular Cone - one whose base is circular. § 565.

(10) Right Circular Cone-one whose vertex lies on the perpendicular to the base drawn from its centre. § 565.

(11) Tangent to a Cylinder or a Cone· a plane containing one element of the lateral surface, and only one; also, a line meeting one element of the surface, and only one. § 549. § 570.

(12) Frustum of a Cone

that portion of a cone intercepted between

its base and a plane parallel to the base, intersecting the lateral surface. § 578.

2. THEOREMS ON THE PROPERTIES OF CYLINDERS.

(1) A plane which contains one element of the lateral surface of a cylinder in general contains also another element, and the section is a parallelogram. § 547.

(2) The section of a right cylinder made by a plane containing an element of the lateral surface is a rectangle. § 548.

(3) The plane determined by a tangent to any circular section of a cylinder and the element of the lateral surface passing through its point of contact is tangent to the cylinder; and conversely, if a plane is tangent to a circular cylinder it intersects the plane of the base in a tangent to the base. § 550.