MISCELLANEOUS RULES. MENSURATION. 1. Mensuration of Surfaces. DEFINITIONS. 1. A point is a small dot; or, mathematically considered, is that which has no parts, being of itself indivisible. 2. A line has length but no breadth. 3. A superficies, or surface, called also area, has length and breadth, but no thickness. 4. A solid has length, breadth, and thickness. 5. A right line is the shortest that can be drawn between two points. 6. The inclination of two lines meeting one another, or the opening between them, is called an angle. 7. If a right line fall upon another right line, so as to incline to neither side, but make the angles on each side equal, then those angles are called right angles, and the line is said to be perpendicular to the other line. 8. An obtuse angle is greater than a right angle. 9. An acute angle is less than a right angle. 10. A circle is a round figure bounded by a single line, in every part equally distant from some point, which is called the centre. 11. The circumference, or periphery of a circle, is the bounding line. 12. The radius of a circle is a line drawn from the centre to the circumference. Therefore, all radii of the same circle are equal. 13. The diameter of a circle is a right line drawn from one side of the circumference to the other, passing through the centre; and it divides the circle into two equal parts, called semi-circles. 14. The circumference of every circle is supposed to be divided into 360 equal parts, called degrees; and each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds; and these into thirds, &c. Hence a semi-circle contains 180 degrees, and a quadrant 90 degrees. 15. An arc of a circle is any part of the circumference. 16. A chord is a right line drawn from one end of an arc to another, and is the measure of the arc. The chord of an arc of 60 degrees is equal in length to the radius of the circle of which the are is a part. 17. The segment of a circle is a part of a circle cut off by a chord. 18. A sector of a circle is a space contained between two radii and an arc less than a semi-circle. 19. Parallel lines are such as are equally distant from each other. 20. A triangle is a figure bounded by three lines. 21. An equilateral triangle has its three sides equal in length to each other. 22. An isocles triangle has two of its sides equal. 23. A scalene triangle has three unequal sides. 27. Acute and obtuse angled triangles, are called oblique angled triangles, or simply oblique triangles; in which the lower side is called the base, and the other two, legs. 28. In a right angled triangle the longest side is called the hypothenuse, and the other two, legs, or base and perpendicular. NOTE. The three angles of every triangle being added together will amount to 180 degrees; consequently the two acute angles of a right angled triangle amount to 90 degrees, the right angle being also 90. 29. The perpendicular height of a triangle is a line drawn from one end of the angles perpendicular to its opposite side. 30. A square is a figure bounded by four equal sides, and containing four right angles. 31. A parallelogram, or oblong square, is a figure bounded by four sides, the opposite ones being equal and the angles right. 32. A rhombus is a figure bounded by four equal sides, but has its angles oblique. 33. A rhomboid is a figure bounded by four sides, the opposite ones being equal, but the angles oblique. 34. The perpendicular height of a rhombus or rhomboides, is a line drawn from one of the angles to its opposite side. 35. A trapezoid is a figure bounded by four sides, two of which are parallel, though of unequal lengths. 36. A trapeze, or trapezium, is a figure bounded by four unequal sides. 37. A diagonal is a line drawn between two opposite angles. 38. Figures which consist of more than four sides are called polygons; if the sides are equal to each other they are called regular poTygons, and are sometimes named from the number of their sides, as pentagon, or hexagon, a figure of five or six sides, &c. If the sides are unequal, they are called irregular polygons. The area of a figure is the space contained between the bounding lines of its surface, without regard to thickness. The area is reckoned so many square inches, square feet, square yards, or square rods, &c. To find the area of a square, or parallelogram. RULE. Multiply the length by the breadth, or perpendicular height, and the product will be the area. 1. How many square rods in a field 28 rods on each side? 28×28=784 rods, Answer. 2. What is the area of a square field, one side of which is 25.35 chains? Answer, 642.622 chains. 3. What is the area of a field 30.5 chains in length, and 24.5 in width? Answer, 747.25 chains. 4. How many square feet in a board 18.8 feet long, and 2.7 feet wide? Answer, 50.76 feet. 5. How many acres in a rectangular piece of ground, 64 rods long and 24 rods wide? Answer, 9 acres. To find the area of a triangle. RULE. Multiply the perpendicular by the base, and one half the product will be the area; or, multiply the base by half the perpendicular height, and the product will be the area. 1. What is the area of a triangle whose base is 20 feet, and whose height 18 feet? 18X20 360-2-180 feet, Answer. 2. What is the area of a triangle whose base is 55 rods, and its height 24.6 rods? Answer, 676.5. 3. How many feet of boards will it take to cover the gable end of a barn, 38 feet wide, the height from the beam to the top being 12.5 feet? Answer, 237.5 feet. When the three sides of a triangle are known, the area may be found by the following RULE. Add together the three sides, and from half their sum subtract each side separately; multiply the half sum and the remainders together continually, and the square root of the product will be the area. 4. What is the area of a triangle whose three sides are 14, 12 and 8 rods? 14+12+8=34-2-17, the half sum: then, 17 17 17 14 12 8 Rem. 3 X5 X 9X17-2295: then /2295-47.9+ rds. 5. The three sides of a triangle are 6, 8 and 10 chains. What is the area? Answer, 24 chains. To find the area of a trapezoid. RULE. Multiply half the sum of the two parallel sides by the perpendicular distance between them, and the product will be the area. 1. What is the area of a piece of land that is 30 chains long, 20 chains wide at one end, and 18 chains at the other? 20+18=38÷2=19, the half sum of the two sides then 19X30-570 chains, Answer. 2. What is the content of a board, 10 feet long, 10 inches wide at one end, and 2 feet 10 inches at the other? Answer, 18 feet. 3. What is the area of a hall, 40 feet long, and at one end 30 feet, and at the other 24 feet wide? Answer, 1080 feet. 4. How many acres in a farm 300 rods long, 80 rods wide at one end, and 60 at the other? Answer, 131 acres, 40 rods. To measure any irregular plane figure. RULE. The whole may be divided into triangles, and measured separately. The sum of the area of the triangles will be the area of the whole. Of the Circle. The circumference of a circle is found by calculation to be about 34 times the diameter, or more accurately by decimals, as 1 is to 3.1416, or as 113 is to 355, so is the diameter to the circumference. Hence, if the diameter is given, to find the circumference, it may be found by multiplying the diameter by 34, or by 3.1416, or as 113 is to 355, so is the diameter to the circumference. 1. What is the circumference of a circle whose diameter is 42 feet? 42x34 132 ft.; or 42×3.1416-131.9472 feet; By reversing the foregoing, the diameter may be found, the circumference being given. 2. If the circumference of a circle be 132 feet, what is the diameter ? 132-34-42 feet, Answer. 3. Suppose the diameter of a circular pond to be 121 rods, what is the circumference? Answer, 380.28+ rods. 4. If the circumference of a circular field be 198 rods, what is the diameter ? Answer, 63 rods. 5. What is the diameter of a tree, whose circumference is 9 feet? Answer, 3 feet. 6. Suppose the circumference of the earth to be 25000. 8528 miles, what is the diameter? Answer, 7958 miles. 7. The diameter of the earth being 7958 miles, what is the circumference? Answer, 25000.8528. To find the area of a circle. RULE. Multiply half the diameter by half the circumference; the product will be the area. 1. What is the area of a circular grove, whose diameter is 147 rods, and circumference 462 rods? Answer, 4622×147÷2=169781 rods. 2. What is the area of a circle whose diameter is 28, and the circumference 88 rods? Answer, 616 rods. 3. How many square rods in a circle whose circumference is 63, and the diameter 20 rods? Answer, 315 rods. The diameter given, to find the area. RULE. Multiply the square of the diameter by .7854, and the product will be the area. 1. What is the area of a circle whose diameter is 28 rods? 28X28X.7854-615.7536 rods, Answer. |