1051. COROLLARY. The lateral area of the frustum of a cone of revolution is equal to the slant height multiplied by the circumference of a section midway between the bases.

PROPOSITION XVI. THEOREM

1052. The volume of the frustum of a circular cone is equal to the sum of its bases and a mean proportional between them multiplied by one third of the altitude of the frustum.

S

Let XS be a frustum of the cone 0-XY, and let B denote the area of the lower base, b the area of the upper base, and a the altitude of the frustum.

To Prove

vol. XS

=

[See proof of § 972.]

a (B + b + √B x b). ļa (B+

1053. COROLLARY. The volume of the frustum of a circular cone is equivalent to the sum of the volumes of three cones whose common altitude is the altitude of the frustum, and whose bases are the upper base, the lower base, and a mean proportional between them.

1054. EXERCISE. The altitude of the frustum of a circular cone is 10 ft. The radii of upper and lower bases are 4 ft. and 9 ft. respectively. Find the volume of the frustum.

1055. EXERCISE.

If the cone in § 1054 is a cone of revolution, find the

lateral area of the frustum.

EXERCISES

1. The volume of a circular cone is a cu. in. and its altitude is b in. Find the radius of its base.

2. If the altitude of a cylinder of revolution is equal to the diameter of its base, its volume is equal to the product of its total surface by one third of the radius of the base.

3. The diameter of a well is 8 ft. Its depth is 10 ft. How many gallons of water will it hold?

4. The slant height of a right circular cone is equal to the diameter of its base. Compare the area of its base with its convex surface.

5. A cone is cut by two parallel planes. Show that the areas of the two sections are to each other as the squares of their distances from the vertex.

6. A cylindrical vessel contains a cu. in. Its height is b in. Find the diameter of its base.

7. Pass a plane parallel to the base of a cone cutting off a section whose area is equal to the base of the cone.

8. Divide a cone into halves by a plane parallel to the base of the cone. 9. The circumference of the base of a right cylinder is a in. The altitude of the cylinder is b in. Find the convex surface and the volume.

10. The radii of the bases of the frustum of a circular cone are 6 in. and 10 in. respectively. Its altitude is 8 in. Find its volume and its convex

surface.

11. The intersection of two planes tangent to a cylinder is parallel to an element.

12. The number of cubic inches in the volume of a certain right cylinder is the same as the number of square inches in its convex surface. Find the radius of its base.

13. The volume of a cone is 400 cu. in. and its altitude is 48 in. Pass a plane, parallel to the base, cutting a section whose area is

sq. in. 14. The altitude of one of two similar cylinders of revolution is 5 times the altitude of the other. Compare their convex surfaces and their volumes. 15. The altitudes of two equivalent right cylinders are as 4 is to 7. If the diameter of the first is 34 ft., what is the diameter of the second?

16. The altitude of a cone of revolution is a ft. and the radius of its base is b ft. Find the dimensions of a similar cone 5 times as large.

17. The lateral area of a cylinder of revolution is equal to the area of a circle whose diameter is a mean proportional between the altitude and the diameter of the base of the cylinder.

18. The volumes of two similar cones of revolution are to each other as 125:216. How do their convex surfaces compare?

19. A right circular cone, whose slant height is equal to the diameter of its base, has the same base and altitude that a right cylinder has. Compare the convex surfaces of the cone and the cylinder.

20. The diameter of a right circular cylinder is 10 ft. and its altitude is 8 ft. What is the edge of an equivalent cube ?

21. The altitude of a cone of revolution is three times the radius of its base. Its lateral area is 200 sq. in. Find its altitude and the radius of its base.

22. Pass a plane, parallel to the base of a cone, cutting off a cone whose volume is one third of the volume of the remaining frustum.

1056. DEFINITIONS.

BOOK IX

A sphere is a solid bounded by a surface, all the points of which are equally distant from a point within called the center.

A radius of a sphere is a straight line drawn from the center to the surface.

A diameter of a sphere is a straight line drawn through the center and terminating in the surface.

A line or a plane is tangent to a sphere if it has one point and only one point in common with the surface of the sphere. Two spheres are tangent to each other when their surfaces have one and only one point in common.

A polyhedron is inscribed in a sphere if all of its vertices are in the surface of the sphere. In this case the sphere is said to be circumscribed about the polyhedron.

A polyhedron is circumscribed about a sphere if all of its faces are tangent to the sphere. In this case, the sphere is said to be inscribed in the polyhedron.

It follows from the definition of a sphere that all radii of the same sphere are equal.

It can be shown that spheres having equal radii are equal; for they can be so placed that their surfaces will coincide. Conversely, equal spheres have equal radii.

A sphere may be generated by revolving a semicircle about its diameter as an axis.

PROPOSITION I. THEOREM

1057. Every section of a sphere made by a plane is a circle.

Let BCE be a section made by a plane cutting the sphere whose center is 0.

To Prove BCE a circle.

Proof. Draw 04 to the plane BCE. From B and C, any two points in the perimeter of BCE, draw BO, CO, BA, and

ሠ/

All points of BCE are equally distant from 4. (?) .. BCE is a circle.

Α

Q.E.D. 1058. DEFINITIONS. Any section made by a plane passing through the center of a sphere is a great circle of the sphere.

A small circle is a section made by a plane that does not pass through the center of the sphere.

A diameter of the sphere that is perpendicular to the plane of a circle of the sphere is the axis of that circle, and the extremities of the diameter are the poles of the circle.

1059. COROLLARY I. center of the circle.

1060. COROLLARY II.

The axis of a circle passes through the

All great circles of a sphere are equal.

1061. COROLLARY III. Circles of a sphere made by planes equally distant from the center of the sphere are equal, and conversely.

1062. COROLLARY IV. Of two unequal circles, the smaller is at the greater distance from the center of the sphere, and conversely. 1063. COROLLARY V. Any two great circles of a sphere bisect each other.

1064. COROLLARY VI. Every great circle of a sphere bisects the sphere and its surface.

1065. COROLLARY VII.

Through two given points on the surface of a sphere, not the extremities of a diameter, the arc of a great circle less than a semicircumference can be drawn, and