Since all the sides of a regular body are equal, it is manifest that the convex surface of any one of them multiplied by the number of sides will give the whole surface. Now, to facilitate the measurement of the surface of any regular body, a Table may be prepared containing the surfaces of the several regular solids, whose linear edges are unity. To construct such a table, we need only multiply the area of one of the sides as is given in Art. 13, by the number of sides. Thus the area of an equilateral triangle, whose edge is 1, is 0.4330127. Consequently the area Of a square, Of a hexagon, =.4330127×4= 1.7320508. =.4330127×6= 2.5980702. Of a regular icosaedron,=.4330127×20=8.6602540. Ex. 1. What is the convex surface of a regular icosae dron, whose edges are each 3 feet ? OPERATION. 3=9, and 9×8.6602540=77.9422860, Ans. NOTE. This question is solved by multiplying the area of one side by the number of sides. Thus, the area of an equilateral triangle, whose edge is 1.4330127. Now, since there are 20 sides, if we multiply this number by 20, the product will be 8.6602540; which, multiplied by the square of one of the sides, gives the whole area as we see in the operation. Ex. 2. Required the surface of a regular dodecaedron whose edges are each 25 inches. Ans. 89.6 sq. ft. Ex. 3. What is the surface of a regular octaedron whose edges are each 76? Ans. 20008.63. PROBLEM II. To find the Solidity of any Regular Solid. ART. 48. Rule.-I. Find the convex surface of the given solid by the previous rule. II. Multiply the surface by of the perpendicular distance from the centre to one of the sides. Or, Multiply the tabular solidity (taken from the following table) in the last column of the table by the cube of the linear edge, and the product will be the solid contents. A Table of Surfaces and Solidities of Regular Bodies, the side being unity, or 1. The above table may be used to great advantage in the measurement of other similar solids. Ex. 1. What is the solidity of an octaedron, whose linear Then, 216x.4714045=101.8233 cubic feet. Ex. 2. What is the solidity of a regular octaedron, whose linear edges are each 32 inches ? Ans. 15447 inches. Ex. 3. What is the solidity of a regular hexaedron, whose linear edges are each 4 feet? Ans. 64 SECTION V. MENSURATION OF THE CYLINDER, CONE AND SPHERE. DEFINITIONS. ART. 49. 1. A Right Cylinder is a solid, having equal and parallel circles for its ends, and is described by the revolution of a rectangle about one of its sides. Thus, EF (fig. I) is a right cylinder. 1 E 2. The Axis of a cylinder is a line passing through the centre, and is perpendicular to the bases; as, EF (fig. 1.) 3. The Height of a cylinder is the perpendicular distance from one base to the plane of the other. 4. If the cylinder be oblique, then the ends are equal and parallel circles, but inclined towards the axis. NOTE-A cylinder may be considered as a prism of an infinite number of sides; for the difference between such a prism and a right cylinder, would be less than any given quantity. 5. A Right Cone is a solid body of a true taper from the base to a point which is called the vertex, and is described by the revolution of a right-angled triangle about one of the sides which contains the right angle; as, AB (fig. 2.) The circle described by the revolving side is called the A 2 base, which is perpendicular to the axis that proceeds from the middle of the base to the vertex, 6. The height of a cone is the fixed side of the triangle by which it is described, or the perpendicular distance from the vertex to the plane of the base; as, AB (fig. 2.) 7. The slant height of a right cone is the distance from the vertex to the circumference of the base. 8. A Frustrum of a cone is what remains after a portion is cut off by a plane, parallel to the base: thus, fig. 3 exhibits the frustrum of a cone. The height of a frustrum is the perpendicular distance between its two parallel ends. 3 9. The slant height of the frustrum of a right cone is the distance between the peripheries of the two ends, measured upon the surface. 10. A Sphere is a solid, terminated by a curved surface, all the points of which are equally distant from a point within, called the centre. A sphere may be described by the revolution of a semi-circle about a diameter. 11. A radius of a sphere is a line drawn from the centre to any part of the surface; as, CA (fig. 4.) 12. The diameter of a sphere is a line drawn through the centre, and terminated at both ends by the surface. All diameters of a sphere are equal to each other, and each is double the radius. 13. A Segment of a sphere is a portion of the sphere cut off by any plane. AB (fig. 5) is a segment of a sphere. This plane is called the base of the segment. A 5 B 14. The height of a segment is the distance from the middle of its base to the convex surface. 15. A Zone is a portion of the surface of a sphere, included between two parallel planes which form its bases. If the bases are equally distant from the centre, it is called the middle zone. 16. The height of a zone is the perpendicular distance between the two planes which form its bases. 18. A Cylindrical Ring is a solid formed by bending a cylinder, as a cylindrical bar of iron, until the two ends meet each other; thus, m,o,n, (fig. 7) is a cylindrical ring. 6 D C 7 B |