285. A cone is a body whose base is a circle and whose convex surface tapers uniformly to a point called the

vertex of the cone.

The altitude of a cone is the shortest distance from the vertex to the center of the base, as AB.

The slant height is the shortest distance from the vertex to the perimeter of the base, as AC.

The convex surface of a cone may be thought of as made up of an infinite number of triangles.

CONE

286. A sphere is a solid bounded by a curved surface, every point of which is equally distant from a point within, called the center.

SPHERE

A straight line passing through the center of a sphere and terminating at both ends in the surface is called its diameter.

One half the diameter of the sphere, or the distance from the center to the surface,

is called its radius.

The greatest distance around a sphere is called the circumference of the sphere.

SURFACES OF SOLIDS

287. Convex surface of a prism or a cylinder. If a prism or a cylinder is 1 in. high, its convex surface contains as many square inches as there are inches in the perimeter of the base. If the height is increased to 2, 3, or any number of inches, the convex surface will be increased in the same proportion. Hence,

To find the convex surface of an upright prism or a cylinder, we multiply the perimeter of the base by the height.

Written Exercise

288. 1. What is the convex surface of a cylinder whose diameter is 3 ft. and whose altitude is 6 ft.?

2. Find the convex surface of a triangular prism whose base measures 8 in. on each side and whose altitude is 15 in.

3. Find the entire surface of a cylinder that is 8 ft. in height and that has a radius of 2 ft.

4. What is the convex surface of a square prism whose sides are each 3.5 ft. and whose altitude is 6 ft.?

289. Convex surface of a pyramid or a cone. The convex surface of a regular pyramid or of a cone is composed of triangles whose bases form the perimeter of the base of the solid, and whose height is the slant height of the solid. Hence,

To find the convex surface of a regular pyramid or of a cone, we multiply the perimeter of the base by the slant height and divide the product by two.

1. What is the convex surface of a cone having a base 4 in. in diameter and a slant height of 9 inches?

2. Find the convex surface of a rectangular pyramid whose base is 15 inches square and whose slant height is 25 inches.

3. At 25 a square yard, what is the cost of painting a church steeple, the base of which is a hexagon 6 ft. on each side, and whose slant height is 75 ft.?

4. How many square yards of canvas are required to make a conical tent 19 ft. in diameter and 12 ft. high?

5. What is the surface of an octagonal pyramid whose base is 13 in. on a side, and whose slant height is 12 in.?

6. Find the convex surface of a cone whose base is 15 ft. in diameter, and whose slant height is 15 ft.

290. Convex surface of a sphere. If we wind a sphere with cord and wind a cylinder whose radius equals the radius of the sphere, and whose height equals the diameter, we find that it

takes as much cord for the cylinder as for the sphere. In the picture, for convenience, only half of each is wound.

The convex surface of a sphere equals the convex surface of a cylinder of the same diameter and height. Hence, it equals the diameter xπd (Art. 287), or the square of the diameter x 3.1416, or πd2.

1. What is the convex surface of a sphere that has a radius of 4 in. ?

2. The earth is a sphere of 8000 mi. diameter. area?

What is its

3. There is a gilded ball on top of a tower. The radius of the ball is 2 ft. How much did it cost to gild it at 10 a square foot?

VOLUMES OF SOLIDS

The same reasoning will give the volume of a cylinder. The volume of a prism or of a cylinder is equal to the product of the area of the base by the altitude.

1. What is the volume of a prism the area of whose base is 16 sq. in., and whose height is 9 in.?

2. Find the volume of a cylinder with a radius of 6 in. and a height of 17 in.

3. How many bushels of wheat will a bin 9 ft. square and 10 ft. high contain?

4. How many gallons of water will a cylindrical reservoir contain, if it is 13 ft. in diameter and 20 ft. high?

5. Find the value, at $.65 a bushel, of corn that will fill a bin the bottom of which measures 9 ft. by 6 ft., and which is 5 ft. deep.

292. Volume of a pyramid. If we construct a hollow prism

and a hollow pyramid of the same base and height, and fill the prism with sand, we find that the pyramid can be filled three times with the same amount. Therefore, we may conclude that:

The volume of a pyramid equals one third the product of the area of the base and height.

293. Volume of a cone. Similarly we may find that:

The volume of a cone equals one third the product of the area of the base and height.

NOTE. It is desirable that these solids should be constructed by the class.

1. What is the volume of a cone, the

diameter of whose base is 5 ft. and whose altitude is 9 ft.?

2. Find the volume of a regular square pyramid, the area of whose base is 64 sq. in. and whose altitude is 17 in.

3. Find the cubical contents of a cone, the radius of whose base is 11 in. and whose slant height is 14 in.

294. Volume of a sphere. If we place this sphere in this hollow cylinder of the same diameter and height, and fill in the

spaces with sand, we find that the sand fills one third of the cylinder after the sphere has been removed. Therefore, we may conclude that:

A sphere equals two thirds the volume of a cylinder of the same diameter and height.

Therefore, since the volume of a cylinder equals Tr2 x 2 r (Arts. 278 and 291), the volume of a sphere equals x πr2 × 2r= 1. Find the volume of a sphere whose radius is 6 ft. 2. What is the volume of a sphere whose diameter is 2.5 ft.?

Written Exercise

295. 1. How many cubic inches are there in a triangular prism, each side of whose base measures 4 in. and whose height is 8 in.? Find also the convex surface and total surface of this prism.

2. How many cubic feet are there in a pyramid 2 ft. high with a base 1 ft. square? Find also the convex surface.

3. A regular hexagonal prism contains how many cubic inches if its height is 6 in. and its base measures 16 square inches?

W. & H. ARITH. IV. -13