OF SUPERFICIAL MENSURATION, OR THE MENSURATION OF PLANE FIGURES, S E C Τ Ι Ο Ν Ι. OF THE AREAS OF RIGHT-LINED AND CIRCULAR FIGURES. T By the menfuration of plane figures is determined the extension of bodies as to length and breadth; such as the quantities of lands, and the works of many artificers. Plane figures, and the surfaces of bodies, are measured by squares; as square inches, or square feet, or square yards, &c; that is, squares whose sides are inches, or feet, or yards, &c. Our least superficial measure is the square inch, other squares being taken from it according to the proportion in the following table. Table of Square Measure. I I Square Inches Sq. Feet 144 S. Yards 1296 S. Poles 9 S. Cla. 39204 272 30 627264 16 6272640 43560 4840 160 IO 40144896002787840013097000 102400 6400 1 4356 484 Acres I To find the Area of a Parallelogram, whether it le a BA B A B * Multiply the length by the height or perpendicular breadth, and the product will be the area. That is, AB X AC = the area. Note. Because the length of a square is equal to its height, its area will be found by multiplying the side by itself.—That is AB X AB or AB’ is the area of the square. E X DEMONSTRATION. For, let ABCD be a rectangle; and let A B its length ab and cd, and its breadth Ad and Bc, be each divided into as many equal parts, as is expressed by the number of times they contain the lineal measuring unit; and let all the opposite points of divifion be connected by right D с lines.-Then, it is evident that, these lines divide the rectangle into a number of squares, each equal to the superficial meafuring unit; and that the number of thefe squares is equal to the number of lineal measuring units in the ' length, as often repeated as there are lineal measuring units in the breadth, or height; that is, equal to the length drawn into the breadth. But the area is equal to the number of squares or fuperficial measuring units; and therefore the area of a rectangle is equal to the product of its length and breadth. Again, a rectangle is equal to an oblique parallelogram of an equal length and perpendicular height, by Euclid I. 36. Therefore the area of every parallelogram is equal to the product of its length and height. 2. E. D. E X AMPLES. 1. What is the area of a parallelogram whofę length is 12.25 chains, and its height 8.5 chains ? 12'25 length 8.5 breadth 35*25 chains ? Note. Four roods are equal to an acre, and therefore 40 perches, or square poles, make a rood. Ex. 2. What is the area of a square whose side is Anf. 124 ac. ir. 1 p. Ex. 3. What is the area of a rectangular board, whose length is 12-5 feet, and breadth 9 inches ? Anf. 9.375 sq. feet. Ex. 4. How many square yards of painting are in a rhombus, or a rhomboid, whose length is 37 feet, and perpendicular breadth 5.25 feet? Anf. 21:58. II. R U LE As radius (viz. fine of 90° or tang. of 45°) Is to the fine of any angle of a parallelogram So is the product of the sides including the angle: To the area of the parallelogram. That B CB CB P * That is, AB X BC X nat. fine of the angle B = the area. Note. Because the angles of a square and rectangle are each 90°, whose fine is i, this rule, for them, is the same as the former. I O'0000000 918 EXAMPLES. 1. What is the area of a rhomboides whose length is 36 feet, slope height 25:5 feet, and one of the less angles 58° ? (Rad.) 1 : 8480481 (nat. sine of 58°) :: 918 (= 25'5 36): 778.5081558 the area. Or, to use the Logarithms. 9-9284205 2.9628427 778.5081 2.8912632 Ex. 2. What is the area of a parallelogram whose angle is 90°, and the including sides 20 and 12.25 chains ? Anf. 245 acres. Ex. 3. What is the area of a rhombus, each of whose sides is 21 feet 3 inches, and each of the less angles 53° 20' ? Ans. 362-208757 feet. Ex. 4. How many acres are in a rhomboides whose less angle is 30°, and the including sides 25-35 and 10•4 chains ? Ans. 13 ac. 29°12 per. RULE DEMONSTRATION. For, having drawn the perpendicular Ap, the area, by the first rule, is aP X BC; but as rad., 1 (s. ZP): S. LB :; AB : AP = S. LB X AB; therefore AP X BC = BC X S. GB X AB, is the area; or 1: s. LB :: AB X BC ; S. LEX AB X BC = the area of the parallelogram. 2. E.D. :: D BD BD B parallelogram make with each other = the area. That is, AB’ in the square, and AB Xcp in the rhombus, is the area. AB XCD X nat. S. LP 2 E XAMPLES. 1. How many square yards of pavement are in a square whose diagonal is 27 feet 6 inches? 27:5 * This rule is common to all quadrilaterals, and is proved at case 2 of prob. 3. |