Ex. 3. The span AB of a roof is 48 feet, and the height EF. 18 feet, standing upon a rectangle plan hipped at each end, I demand the length of the common rafters, and likewise the hip rafters. +576 +324 9:00(30 9 50) 00 00 D C Draw Geometrical construction. AD and BC perpendicular to AB, and make AD equal to AE or EB, and draw DC parallel to AB: join ED and EC; produce CE to G, making EG equal to EF, and join GD; then will GD be the length of the hip rafter. Therefore, the length of the common rafters are 30 feet each. But because the hip rafters are the hypothenuses of right-angled triangles, having a common rafter for one of the perpendicular legs, and the other leg being equal to half the width of the roof. Case 2.-The hypothenuse and one of the legs being given, to find the other leg. From the square of the hypothenuse, take the square of the given leg, and the square root of the remainder will be equal to the other leg. Ex. 1. In the right-angled triangle ABC, the base AB being 40, and the hypothenuse AC 50, I demand the perpendicular BC. Ex. 2. The width AB of a roof being 36 feet, and the length of a rafter AC or BC 25 feet, I demand the height of the roof. 17100 Prob. 5. To find the area of a trapezium. Multiply the diagonal by half the sum of the two perpendiculars, falling upon it from the opposite angles, and the product will be the area. Er. What is the area of a trapezium ABCD, the diagonal AC being 36 feet, the perpendicular DE 16 feet, and BF 12 feet? 16 +12 Prob. 6. To find the area of a trapezoid. Multiply the half sum of the parallel sides by the perpendicular distance between them, and the product will be the area. Ex. 3. What is the area of a board or plank in the form of a trapezoid, being 1f. 71. at one end, 2f. 31. at the other end, and 8f. 61. long? Prob. 7. To find the area of any regular polygon. Multiply half the perimeter of the figure by the perpendicular, falling from its centre upon one of the sides, and the product will be the area of the polygon. Ex. 1. Requireth the area of a regular pentagon ABCD, whose side AB, or BC, &c. is 6 feet, and the perpendicular FG 4 feet. Er. 2. How many feet of ground does an hexangular building cover, each side of the base being 8f. 3i. and the perpendicular 7 feet? 8 3 2)49 6 24 9 half the sum of the sides. f.173 3 Prob. 8. To find the area of a polygon, when the side only is given. Multiply the square of the given side of the polygon by that number which stands opposite to its name in the following Table, and the product will be the area. In the above Table, those multipliers marked with the sign +, are too small; on the contrary, those marked, are too great: I have only given this Table to five places of decimals, being exact enough for most practical purposes. Er. 1. Requireth the area of a pentagon, each side being 14 feet. Prob. 9. The diameter of a circle being given, to find the circumference; or the circumference being given, to find the diameter. Method 1. As 7 is to 22, so is the diameter to the circumference nearly; or, as 22 is to 7, so is the circumference to the diameter nearly. Ex. 1. What is the circumference of a circle, whose diameter is 12 feet? 7 22: 12 7)26 371. 81. Ex. 2. What is the diameter of a circle, whose circumference is 03 feet? The most practical method to find the circumference of a circle from its diameter, is the following: Multiply the diameter by 3: add part of the diameter to the product, the sum will be the circumference, the same as in Method 1. Er. 1. What is the circumference of a circle, whose diameter is 4f. 6i.? Ex. 2. What is the circumference of a semicircular vault, whose diameter is 16f. 5in. ? in. 16 5 49 3 2 4 1 2)51 7 1 feet 25 9 6 answer nearly. |