Ex. Let the sides of the triangle be 3, 6, 7, yards respectively; to find the area of the triangle. 3+6+7=16, .. S=8, S-AB-8-3=5, S-AC-8-6=2, S-BC-8-7=1; .. area=√8×5×2×1=√80=8′95 square yards nearly. 231. When the length and breadth of a rectangular figure are given, we have seen (223) that the area is found by multiplying together these two dimensions, taking care that they are both measured by the same unit. These dimensions are often expressed in feet; or in feet and inches; or in feet, inches, and fractional parts of an inch. If these fractional parts are halves, or quarters, it is usual to express them in twelfths. We subjoin examples. 1st. Let the dimensions be expressed solely in feet, as 5 ft. by 3 ft. Then the area is equal to 5×3, or 15 square feet. 2nd. Let the dimensions be in feet and inches; as 5 ft. 3 in. by 4 ft. 6 in. Here it must be observed, that in the required multiplication, we have these results, viz. 1 foot x 1 inch = an area of 12 sq. inches, Hence, in the subjoined operation the work is performed as in compound multiplication, and the units in each denomination are converted into the next higher by dividing by 12; so that the result is 23 sq. ft., 7 areas of 12 sq. inches and 6 sq. inches; or 23 sq. ft. 90 sq. inches. 5.3 4.6 2.7.6 21.0 23.7.6 3rd. Let the dimensions be in feet, inches, and twelfths of an inch; as in a rectangle 5 ft. 4 in. 3 twelfths, by 2 ft. 6 in. 9 twelfths. We have now to observe that 1 foot × 1 inch = 12 in. x in. = 1 sq. in. 1 in. x in. in. x sq. in., Hence, in the accompanying operation, the product is obtained by the same mode as in the last example, and is 13 sq. ft., 8 areas of 12 sq. in., 7 sq. in., 8 areas each sq. in., and 3 each sq. in. The second and third products when reduced to one name amount to 103 sq. in.; and the fourth and fifth to 14 of a square 5.4.3 2.6.9 4.0.2.3 2.8.1.6 10.8.6 13.8.7.8.3 inch: : or, the whole = 13 sq. ft. 1031 sq. in. 4th. If the fractional parts of an inch cannot be converted into twelfths, it is better to bring the whole to feet and decimal or fractional parts of a foot, or of an inch. Thus 7 ft. 2 in. would be better converted into 7 ft., or 86 in., or 86-2 in. If the resulting decimal would be recurring, a vulgar fraction is preferable. Ex. Find the rectangular area, whereof the length and breadth are 3 ft. 54 in., and 2 ft. 6 in. respectively. The product=414 x 308 sq. inches = 41×303 288 272 X -X 1 7 9 144 =848 sq. ft. If the dimensions be 1 ft. 10.6 in. and 0.275 in. the product=22.6 × 0.275 sq. in. = = 6.215 sq. in. 04316 sq. ft. nearly. PART III. 2 QUESTIONS AND EXERCISES F. (1) What is meant by the length of a line in Mensuration? (2) What is meant by choosing a unit of measurement? (3) Shew by examples that there is an advantage in selecting particular units of measurement. (4) How many different kinds of units can be employed in Mensuration? Are they all employed in the preceding section. (5) If, in ascertaining the length of a line, according to Question (1), you find the result to be a fraction, how do you interpret that result? (6) Describe the common modes of measuring (1) short straight lines, (2) long straight lines. (7) Shew how to measure a proposed crooked, or curved, line. (8) If one foot is taken as the unit of length, what are the numbers which represent the several lengths of 3 in., 2 in., 2 ft. 3 in., 7 yds., all in terms of that unit? (9) What is the mode of ascertaining the number of units in any proposed square? (10) What is the reverse process, viz. when the number of units in the square is given, to find the length of one side? (11) Shew how the Table commonly called "Square Measure" is constructed. (12) What is the most ready mode of finding the area of a triangle? (13) Write down the expression for the area of a triangle, in terms of its base and perpendicular height; and shew that if the base of a triangle be 625 poles and its height 1.2 poles, its area is 11.34375 sq. yds. (14) Define a rectangle what other name is given (15) Can we always express the area of a proposed rectangular figure in terms of units arbitrarily taken; if not, why not? (16) Describe the required measurements and calculations for finding the area of a rectangle. (17) Suppose a parallelogram be not rectangular, what are the required measurements for finding its area? (18) Give an example of two lines which are incommensurable, and shew that they are so. (19) Deduce from the measurement of a parallelogram the measurements required for finding the area of a triangle. (20) The area of a parallelogram is 375 sq. yds.; its base is 75 yds.; what is the perpendicular height? Ans. yd. (21) The area of a triangle is 3.525 acres, and the perpendicular from the vertex of one angle on the opposite side is 1936 yds.; find the length of that side. Ans. 176.25 yds. (22) Deduce from question (19) the mode of measuring a plane surface bounded by any given number of straight lines. (23) How may the perimeter and area of a regular polygon be ascertained? (24) In the following irregular four-sided figures, given the diagonal, and the perpendiculars upon it from the opposite angles, in each case, find the areas. Diagonal. (1) 27.6 ft. (2) 35 ft. 2 in. (1) Ans. 343 62 sq. ft. Perps. 13′2 ft., and 11·7 ft. 17 ft. 2 in., and 16 ft. 3 in. 35.7 ft., and 45'9 ft. (2) Ans. 588 sq. ft. 391 sq. in (3) Ans. 454 sq. yds. 4-608 sq. ft. sq.i (25) In the following five-sided figures, given the diagonals and three perpendiculars from opposite angles, in each case, find the areas. Note. The first two perpendiculars are upon the first diagonals. (1) Ans. 752.5. (2) Ans. 155-76. (3) Ans. 142.362. (26) The perimeter of a regular hexagon is 75 ft.; find the radius of the circumscribing circle. Ans. 14 inches. (27) Find the areas corresponding to the following lengths and breadths of rectangular figures: (3) Ans. 61 sq. ft. 115 sq. in. (4) Ans. 372 sq. ft. 231 sq. in. (28) The following dimensions of rectangles are expressed in feet and decimal parts of a foot; or in inches and decimal parts of an inch. Find the areas. (1) Ans. 24.525 sq. ft. (2) Ans. 75 sq. ft. 32 sq. in. (3) Ans. 095625 sq. in. (4) Ans. 2 sq. ft. 23.2368 sq. in. (29) Find the areas of the triangles, whereof one side and the perpendicular thereon from the vertex of the opposite angle are respectively as follows: |