SOLIDS 586. Anything that can be measured in three directions is said to have three dimensions,-length, breadth, and thickness,— and is called a solid. 587. The surfaces that bound a solid are called its faces and their intersections its edges. 588. The number of cubic units that any solid contains is called its volume. 589. What name (§ 247) is given to a solid having six rectangular faces? to a rectangular solid whose faces are equal squares? (§ 248) 590. A solid whose sides are parallelograms and whose two ends are equal polygons, parallel to each other, is called a prism. Prisms are named from the shape of their bases, triangular, square, rectangular, pentagonal, etc. TRIANGULAR SQUARE PRISM PRISM 591. The perpendicular distance between the bases of a prism is called its altitude. 592. The parallelograms taken together form the convex surface of the prism. 593. A solid bounded by a uniformly curved surface and having for its bases circles that are parallel to each other is called a circular cylinder. CYLINDER There are other kinds of cylinders, but in this book "cylinder" means "circular cylinder." A cylinder may be thought of as a prism whose bases are circles and whose convex surface is made up of an infinite number of infinitely narrow parallelograms. 594. A solid whose base is a polygon and whose faces are triangles meeting at a point is called a pyramid. The triangles form the convex surface of the pyramid, and the point where they meet is called the vertex. Pyramids, like prisms, are named from their bases, as triangular, square, hexagonal, etc. 595. The perpendicular distance, as AB, from the vertex to the base of a pyramid is called its PYRAMID altitude. 596. The altitude, as AC, of one of the triangles of a pyramid is called the slant height of the pyramid. 597. A solid whose base is a circle and whose surface tapers uniformly to a point, called the vertex, is a circular cone. In this book "cone" means "circular cone." The vertex of a cone is sometimes called its apex. A cone may be thought of as a pyramid whose base is a circle and whose convex surface is made up of an infinite number of infinitely narrow triangles. CONE The altitude and slant height of a cone correspond to the altitude and slant height of a pyramid. 598. A solid bounded by a curved surface every point of which is equally distant from a point within, called the center, is a sphere. 599. A straight line passing through the center of a sphere and terminating. at both ends in the surface is called its diameter. SPHERE 600. One half the diameter of a sphere, or the distance from the center to the surface, is called its radius. 601. A circle of a sphere whose plane passes through the center is called a great circle of the sphere. 602. A great circle divides a sphere into two equal parts called hemispheres. The circle is the base of each hemisphere. 603. The circumference of a great circle of a sphere is called the circumference of the sphere. The circumference of a sphere is the greatest distance around it. Surfaces of Solids 604. To find the convex surface of a prism or a cylinder. 1. How many square inches are there in the convex surface of a prism or a cylinder 1 inch high, if the perimeter of its base is 6 inches? 2. What is the area of the convex surface, if the altitude of the solid is 2 inches? 3 inches? 4 inches? 3. How, then, may you find the area of the convex surface of a prism or a cylinder? 4 3 605. The convex surface of a prism or a cylinder is equal to the product of its altitude and the perimeter of its base. 606. 1. What is the convex surface of a cylinder whose diameter is 2 feet and whose height is 5 feet? 2. Find the convex surface of a triangular prism whose base is 2 centimeters on each side and whose altitude is 4 centimeters. 3. What is the convex surface of a square prism whose sides are each 21 feet and whose altitude is 6 feet? 4. Find the entire surface (convex surface and surface of the two bases) of a cylinder that is 8 feet in height and has a base 3 feet in diameter. THIRD PROG. AR. - 21 607. To find the convex surface of a pyramid or a cone. 1. You have learned that the convex surface of a pyramid is composed of triangles, and that the convex surface of a cone may also be assumed to be made up of an infinite number of triangles. The bases of these triangles form the perimeter of the base of the solid, and their altitude is the slant height of the solid. 2. How, then, may you find the area of the convex surface of a pyramid or a cone? 608. The convex surface of a pyramid or a cone is equal to one half the product of its slant height and the perimeter of its base. 609. 1. What is the convex surface of a rectangular pyramid whose base is 6 feet square, and whose slant height is 5 feet? 2. Find the convex surface of a cone having a base 5 centimeters in diameter and a slant height of 3 decimeters. 3. What is the convex surface of a cone whose base is 20 feet in diameter, and whose slant height is 20 feet? 4. At 30 per square yard, what is the cost of painting a church steeple, the base of which is an octagon 6 feet on each side, and whose slant height is 80 feet? 5. How many feet of convex surface are there on a cone, the base diameter of which is 6 feet, and whose slant height is 91 feet? 6. How many feet of convex surface are there on a pyramid whose base is 10 feet square, and whose slant height is 20 feet? 7. How many feet of convex surface are there on a cone whose base is 8 feet in diameter, and whose slant height is 6 feet? 610. To find the convex surface of a sphere. 1. The length of a waxed cord sufficient to cover the convex surface of a hemisphere, when carefully wound as shown in the picture, is just twice the length of cord required to cover the base of the hemisphere; that is, the area of the convex surface of a hemisphere is twice the area of its base. 2. Then how many times the area of a great circle of a sphere is the convex surface of the whole sphere? 3. It may be proved also by geometry that the convex surface of a sphere is equal to 4 great circles. 4. How does the radius of a great circle of a sphere compare with the radius of the sphere? What formula (§ 583) expresses the area of a circle? Then, what formula expresses the area of 4 great circles, or the convex surface of a sphere ? 611. The convex surface of a sphere is equal to 4 great circles, or to 4 πr2. 612. Denote the diameter by d. d2, the convex surface of a sphere is diameter times π, or πd 2. Then, since 4 r2 = (2 r)2= equal to the square of the 613. 1. What is the convex surface of a baseball 3 inches in diameter ? 2. Find the convex surface of a rubber ball having a radius of 12 centimeters. 3. What is the convex surface of a spherical cannon ball 8 inches in diameter? 4. Find the cost, at 12 cents a square foot, of gilding a sphere 28 inches in diameter. |