4. Required the side of an equilateral triangle that can be inscribed in a circle 251.33 feet in circumference. 5. How much less is the area of an equilateral triangle that can be inscribed in a circle 100 feet in diameter, than the area of the circle itself? Ans. 4606.4 sq. ft. THE ELLIPSE. 655. An ELLIPSE is a plane figure bounded by a curve, from any point of which the sum of the distances to two fixed points is equal to a straight line drawn through those two points, and terminated both ways by the curve. Thus A DBC is an ellipse. The two fixed points G and H are called the foci. The longest diameter, AB, of the ellipse is called its major or transverse axis, and its shortest diameter, CD, is called its minor or conjugate axis. C G H B D 656. The AREA of an ellipse is a mean proportional between the areas of two circles whose diameters are the two axes of the ellipse. This, however, can only be well demonstrated by means of Analytical Geometry, a branch of the mathematics with which the learner here is not supposed to be acquainted. PROBLEM XXX. 657. To find the area of an ELLIPSE, the major and minor axes being given. Multiply the axes together, and their product by 0.7854, and the result will be the area. For A B2 × 0.7854 expresses the area of a circle whose diameter is A B, and C D2 × 0.7854 expresses the area of a circle whose diameter is CD; and the product of these two areas is equal to A B2 X CD2 X 0.78542, which is equal to the square of AB X CD × 0.7854; hence, AB XCD X 0.7854 is a mean proportional between the areas of the two circles whose diameters are A B and CD (Prop. IV. Bk. II.); consequently it measures the area of an ellipse whose axes are AB and CD (Art. 656). EXAMPLES. 1. Required the area of an ellipse, of which the major axis is 60 feet, and the minor axis 40 feet. 60 X 40 X 0.78541884.96 sq. ft., the area required. 2. What is the area of an ellipse whose axes are 75 and 35 feet? 3. Required the area of an ellipse whose axes are 526 and 354 inches. Ans. 112 yd. 7 ft. 84.62 in. 4. How many acres in an elliptical pond whose semiaxes are 436 and 254 feet? Ans. 7 A. 3 R. 37 P. 27 yd. 7 ft. BOOK XII. APPLICATIONS OF GEOMETRY TO THE MENSURATION OF SOLIDS. DEFINITIONS. 658. MENSURATION OF SOLIDS, or VOLUMES, is the process of determining their contents. The SUPERFICIAL CONTENTS of a body is its quantity of surface. The SOLID CONTENTS of a body is its measured magnitude, volume, or solidity. 659. The UNIT OF VOLUME, or SOLIDITY, is a cube, whose faces are each a superficial unit of the surface of the body, and whose edges are each a linear unit of its linear dimensions. 660. TABLE OF SOLID MEASURES. 1728 Cubic Inches make 1 Cubic Foot 661. To find the surface of a RIGHT PRISM. Multiply the perimeter of the base by the altitude, and the product will be the CONVEX surface (Prop. I. Bk. VIII.). To this add the areas of the two bases, and the result will be the ENTIRE surface. EXAMPLES. 1. Required the entire surface of a pentangular prism, having each side of its base, A B C D E, equal to 2 feet, and its altitude, A F, equal to 5 feet. 2. The altitude of a hexangular prism is 12 feet, two of its faces are each 2 feet wide, three are each 2 feet wide, and the remaining face is 9 inches wide; what is the convex surface of the prism? 3. Required the entire surface of a cube, the length of each edge being 25 feet. 4. Required, in square yards, the wall surface of a rectangular room, whose height is 20 feet, width 30 feet, and length 50 feet. Ans. 355 sq. yd. PROBLEM II. 662. To find the solidity of a PRISM. Multiply the area of its base by its altitude, and the product will be its solidity (Prop. XIII. Bk. VIII.). EXAMPLES. 1. Required the solidity of a pentangular prism, having each side of its base equal to 2 feet, and its altitude equal to 5 feet. 22 X 1.72048 = 6.88192; 6.88192 x 534.40960 cubic [feet, the solidity required. 2. Required the solidity of a triangular prism, whose length is 10 feet, and the three sides of whose base are 3, 4, and 5 feet. Ans. 60. 3. A slab of marble is 8 feet long, 3 feet wide, and 6 inches thick; required its solidity. 4. There is a cistern in the form of a cube, whose edge is 10 feet; what is its capacity in liquid gallons? Ans. 7480.519 gallons. 5. Required the solid contents of a quadrilateral prism, the length being 19 feet, the sides of the base 43, 54, 62, and 38, and the diagonal between the first and second sides, 70 inches. Ans. 306.047 cu. ft. 6. How many cords in a range of wood cut 4 feet long, the range being 4 feet 6 inches high and 160 feet long? PROBLEM III. 663. To find the surface of a RIGHT PYRAMID. Multiply the perimeter of the base by half its slant height, and the product will be the CONVEX surface (Prop. XV. Bk. VIII.). To this add the area of the base, and the result will be the ENTIRE surface. 664. Scholium. The surface of an oblique pyramid is found by taking the sum of the areas of its several faces. EXAMPLES. 1. Required the convex surface of a pentangular pyramid, ABCDE-S, each side of whose base, A B C DE, is 5 feet, and whose slant height, SM, is 20 feet. S A D 25; 25 × 20 2. What is the entire surface of a triangular pyramid, of which the slant height is 18 feet, and each side of the base 42 inches? Ans. 99.804 sq. ft. 3. Required the convex surface of a triangular pyramid, the slant height being 20 feet, and each side of the base 3 feet. 4. What is the entire surface of a quadrangular pyramid, the sides of the base being 40 and 30 inches, and the slant height upon the greater side 20.04, and upon the less side 20.07 feet? Ans. 125.308 ft. |