MENSURATION, § 196. Is the art of measuring superficies or solids. DEFINITIONS. 1. A point is that which has position but not magnitude. 2. A line has length but not breadth, and its bounds or extremes are points. 3. A right line is the shortest line that can be drawn between any two points. 4. A superficies or surface is that which has length and breadth but not thickness. 5. A figure is space bounded by one or more lines. 6. The area or superficial content of any figure, is the space contained within the line or lines which bound it. 7. Parallel lines are those which are in the same plane, and which being produced both ways do not meet. 8. An angle is the inclination or opening of two lines, having different directions and meeting in a point. 9. One line is perpendicular to another when it makes the angles on both sides equal to each other. 10. A right angle is that which is formed by two lines that are perpendicular to each other, and contains 90°. 11. An acute angle is less than a right angle. 12. An obtuse angle is greater than a right angle. 13. A triangle is a figure bounded by three right lines, and has three angles. 14. An equilateral triangle has its three sides all equal. 15. An isosceles triangle has only two of its sides equal. 16. A scalene triangle has three sides all unequal. 17. A right-angled triangle has one right angle; the side opposite which is called the hypothenuse, and the other two sides the legs, or the base and perpendicular. 18. A figure of four sides is called a quadrangle, or a quadrilateral. 19. A square is a quadrilateral, whose sides are all equal, and its angles all right angles. 20. A rhombus is a quadrilateral, whose sides are all equal. but its angles not right angles. 21. A parallelogram is a quadrilateral, whose opposite sides are parallel. 22. A rectangle is a parallelogram, whose angles are all right angles. 23. A rhomboid is a parallelogram, whose angles are not right angles. 24. A trapezium is a quadrilateral, which has not its opposite sides parallel. 25. A trapezoid is a quadrilateral, with two sides parallel. 26. A diagonal is a line joining any two opposite angles of a quadrilateral. 27. Figures having more than four sides are called polygons. 28. Polygons having five sides, are called pentagons; those of six sides, hexagons; those of seven, heptagons; &c. 29. The base of any figure is that side on which it is supposed to stand, and the altitude is the perpendicular falling upon it from the opposite angle. 30. A circle is a plane figure bounded by a curve line which is equally distant from a point within, called the centre. The boundary line is called the circumference. 31. An arc is any portion of the circumference. 32. A semicircle is half, and a quadrant, quarter of a circle. 33. The radius of a circle is a line drawn from the centre to the circumference. 34. The diameter of a circle, is a right line drawn through the centre, terminated both ways by the circumference. 35. A solid is that which has length, breadth, and thickness, and its bounds or extremes are superficies. 36. A cube is a solid contained by six equal square sides. 37. A parallelopipedon is a solid contained by six quadrilateral planes, every opposite two of which are equal and parallel. 38. A prism is a solid whose ends are two equal parallel and similar plane figures, and whose sides are parallelograms. 39. A cylinder is a solid described by the revolution of a rectangle about one of its sides, which remains fixed. 40. A cone is a solid formed by the revolution of a rightangled triangle about one of its legs, which is fixed. 41. If a cone be cut by a plane passing obliquely through its two slant sides, the section will be an ellipsis. 42. The longest diameter of the ellipsis is called the tranverse, and the shortest the conjugate. 43. A pyramid is a solid whose sides are all triangles meeting in a point at the vertex, and the base any plane figure whatever. 44. A sphere or globe is a solid described by the revolution of a semicircle about its diameter, which remains fixed 45. The diameter of a sphere is a right line passing through the centre, and terminated on each side by the convex surface. 46. The areas of circles are to each other as the squares of their diameters, radii, or circumferences. The areas of similar figures are to each other, as the squares of their like sides. 47. The surfaces of all similar solids are to each other as the squares of their like dimensions; such as diameters, circumferences, like linear sides, &c. And their solidities, as the cubes of those dimensions. PROBLEM I. To find the area of a square. EXAMPLES. 1. What is the area of a board whose side is 19 inches? 19×19=361 area required. 2. How many acres in a square field, whose side is 50 rods? 3. What is the area of a square, whose side is 4 ft. 2 in. ? PROBLEM II. The area of a square being given to find the length of the side. RULE. Extract the square root of the area. EXAMPLES. 1. The area of a square is 729 yards: what is the side? ✓729-27-side required. 2. What is the side of a square floor containing 2025 sq. ft. ? 3. What is the side of a square, whose area is 6A. 1R.? PROBLEM III. The diagonal of a square being given, tɔ find the area. RULE. Divide the square of the diagonal by 2. EXAMPLES. 1. The diagonal of a square is 8 chains: what is the area? (8x8)+2=32 chains-3 a. 2 r. 2. The diagonal of a square is 16 feet, what is the area? 3. The diagonal of a square is 12 feet, what is the area? PROBLEM IV. The area of a square given, to find the diagonal. RULE. Extract the square root of double the area. EXAMPLES. 1. The area of a square piece of land is 28.8 acres, what is the diagonal in chains? ✓(288x2)=√576-24 chains 2. The area of a square is 578 feet, what is the diagonal? 3. The area of a square is 128 yards, what is the diagonal ? PROBLEM V. The diagonal of a square given, to find the side. RULE. Find the square root of half the square of the diagonal. EXAMPLES. 1. The diagonal of a square is 12yds., what is the side ? √((12×12)÷2)=√✔/72=8.485. 2. The diagonal of a square is 36 feet, what is the side? 3. The diagonal of a square is 48 chains, what is the side? PROBLEM VI. To cut off a given area from a square, parallel to either side. RULE. Divide the given area by the length of the side, the quotient will be the length on the other side. EXAMPLES. 1. What length must be cut from a square whose sides are 25 chains, to leave 30 acres at the end? 300 ch. 25-12 chains, length required. 2. The sides of a square are 17 feet, what must be the length of the other side to leave 153 square feet? 3. The side of a square being 42 rods, what must be the length of the other side, to leave 4 acres ? PROBLEM VII. The length and breadth of a rectangle being given, to find the area. RULE. Multiply the length by the breadth. EXAMPLES. 1. What is the area of a rectangle whose length is 14.5 feet, and breadth 11.6 feet? 14.5×11.6=168.2 feet. 2. What is the area of a rectangle whose length is 117.8 yards, and breadth 204.7 yards? 3. What is the area of a rectangular field whose length is 87 rods, and breadth 56 rods? |
