E. I. If the femi-diameter A B, of a circle, be 24,5, the arch line. 45,6, the chord upon which the triangle is formed 30,5, and the perpendicular of the triangle 16,2; what is the area of the fegment? E. 2. 395 2440 247,05= Arch of the triangle. What is the area of a fegment whofe radius is 11,64, arch line 48, the chord upon which the triangle is formed 20,5, and the perpendicular of the triangle 5,53? RULE 2. First, add the fquare of half the chord of the fegment to the fquare of its height, and multiply the square root of the sum by 4. 2. To of the number laft found, add the whole chord of the fegment, this fum multiplied by of the height, will give the area. E. 1. If the chord of a fegment be zo, and its height or verfed fine 5, what is the area of the segment? 2R Anf. 69,812 Area of the fegment. PROB 34,906 of the height PROB. 13. To find the area of an ellipfis, or oval. RULE. Multiply the tranfverfe diameter by the conjugate, then multiply that product by ,7854, this laft product is the area of the oval. E. 1. What is the area of an ellipfis, whofe tranfverfe diameter T is 22, and conjugate C 16? E. 2. If the axis of an ellipfis be 36 and 26, what is the area? RULE. PROB. 14. To find the area of a parabola. Multiply the bafe, or greatest ordinate, by the height, or abfciffa; and of the product will be the area. 307 What is the area of a parabola, whose base or greatest ordinate E. 2. 24, and the abfciffa 8? the area, anfwer. =1406,4583, PROB. 15. To find the area of an hyperbola. RULE. Multiply the base, or greatest ordinate, by the height, or abfciffa, and of the product will be the area, nearly. E. 1. What is the area of an hyperbola, whofe base, or greateft ordinate O is 24, and the abfciffa or height X, 10? 24 10 240 5 8)1200 Anf. 150 The area. E. 2. Required the area of an hyperbola, whose base, or greatest ordinate, is 36, and the perpendicular height, or abfciffa, 12 ? 36 12 432 5 8)2160 Answer 270 The area. Note. The above rule is only an approximation, but will ferve very well for common purposes. PROB. 16. To find the area of a spherical triangle. RULE. From the fum of the three angles, fubtract 180 degrees; multiply the fuperficies of the whole fphere, or globe, by the remainder, this product divide by 720; the quotient is the area of the triangle. E. 1. Suppofe the angle at A = 36°; at B 148°; at C 32°, and the diameter of the globe 29; what is the area of the triangle ABC? 2R 2 Fift Note. By this problem you may find the number of miles or acres contained in the whole, or any part of the furface of the globe. PROB. 17. To find the areas of lunes, or the spaces included between the interfe&ting arches of tapo eccentric circles. (See Plate 1, Fig. 31.) RULE. Find the areas of the two fegments from which the lune is formed, and their difference will be the area required. EXAMPLE. Suppose the length of the chord A C is 40, the height EF 10, and ED 4; what is the area of the lune AFCDA? First 400 Square of half A C Square of half AC And √ 416 Again 400 16= = 100 = Square of E F 416 20,396 69,81 67194 of the height 107,5104 Area of the fegment 279,24 Area of the fegment [ACDA, [ACFA. 279,24 107,5104 Anfw. 171,7296 The area of the lune required. *The furface of a Sphere may be found by Problem 9, Sect. 70% LXVIII. LXVIII. MENSURATION F SOLIDS. 'EACHETH how to meafure all folid bodies, which confift of length, breadth, and thickness. TEA PROB. I, To find the folidity of a cube. RULE. Multiply the fide of the cube into itself, and that product again by the fide, and it will give the folidity. E. I. The fide A B of the cube is 6,5, what is the folidity ? Note. If the answer be in inches, you must divide by 1728 (the folid inches in a foot) to bring them into feet. PROB. 2. To find the folidity of a parallelopipedon, RULE. Multiply the length by the breadth, and that product again by the depth or altitude, and it will give the folidity. |