Mensuration of Plane Surfaces. 1. To find the hypothenuse of a right-angled triangle, when the other sides are known,—— Square the sides, add the squares together, and extract the square root. 2. To find one side, the hypothenuse and remaining side being given, Square the hypothenuse and the side; subtract the latter product from the former, and take the square-root of the remainder ; or, Multiply the sum of the hypothenuse and side by their difference, and take the square-root of the product. 3. To find the area of a triangle, when the base and altitude are given,— Multiply the base by the altitude and divide by 2. 4. To find the circumference of a circle when the diameter is given, 5. To find the area of a circle when the radius is given,— Multiply the square of the radius by 22 7 6. To find the area of a sector of a circle, the radius and arc being given, Take half the product of the arc and radius, 7. To find the area of a square,— (1) When the side is given,— Square the side. (2) When the diagonal is given,— Square the diagonal and divide by 2. 8. To find the area of a rectangle when the sides are given,— Multiply the length by the breadth. 9. To find the area of a rhombus when the two diagonals are given, Take half the product of the diagonals. 10. To find the area of a trapezoid, the parallel sides and the breadth being given,— Multiply the sides of the parallel sides by the breadth, and divide by 2. II. To find the area of an irregular figure bounded by straight lines,― (1) Divide the figure into triangles, and add their areas together. (2) Take a base-line and draw perpendiculars from the angles of the figure to it; and find the areas of the trapezoids and triangles thus formed :—the area of the figure = areas of the trapezoids + areas of the triangles within the figure—areas of triangles without the figure. 12. To find the area of a rectilineal figure, when one side of the figure, and the area and a like side of a figure similar to it are given,— Square the sides; the area required is to the area given as the squares of the like sides. 13. To find the area of a triangle when the three sides are given, Subtract each side separately from half the sum of the sides: multiply together the three remainders and the half sum, and find the square root of the product. INDEX TO PLANE GEOMETRY. The numbers here given refer to the pages. Arithmetical questions, 25, 40, 76, 93, 120, 139, 160, 186, 198, 230, 259, 265, 266. Auxiliary curve, 210. Bisection of lines, 39. Concentric circumferences, 161. Continuity, law of, 146. Definitions of terms in Plane Geometry,- Altitude of triangles, 191. Angles, 9-11. Area, 76. Circle and its parts, 25. Departure, 20. Equivalent figures, 76. Geometrical locus, 15. Homologous lines, 187. Parallelograms and other quadrilaterals and their parts, 60-62. Equivalence and similarity contrasted, 196. GEOMETRY APPLIED IN PRACTICAL CONSTRUCTIONS :- Circles and circumferences fulfilling given conditions, 26- Circles touching and cutting one another, 164-172. Division of angles, 64, 126. Division of polygons, 238-242. Equal angles, 37. Involute to circle, 218, 219. Ovals and elliptical figures, 175-178. Parallelograms, 63. Parallels, 52-57. Perpendiculars, 38, 39, 43. Polygons in and about circles, and vice versa, 202-210. Proportionals, 107-109, 134. Proportional figures, 250-252. Rectangles, 78, 195. Regular polygons, 232-237. Regular stars, 223. Secants fulfilling various conditions, 124-132. Similar polygons, 243, 244. Squares, 83. Star polygons, 222; by extension, 225; by reduction, 224. Tangents, 141–145. Triangles, 45-47. Horizontal plane, 3. Lozenge (or rhombus), 65. Parallelogram of forces, 69. Froblems for solution, see "Theorems and Problems for Solu- Proportion of numbers, 96-100. Propositions, see "Recapitulation," 267. Quadrant, 35. Questions for examination, 23, 39, 50, 72, 91, 118, 137, 156, 182, 196, 228, 257. Recapitulation of important propositions, 267. Rise of an arc, 31. Rules for mensuration of plane surfaces, 272. Symmetry, axis of, 15. Symmetry, centre of, 65. TECHNICAL APPLICATIONS OF GEOMETRICAL PRINCIPLES:- GRADUATION OF CIRCLES AND VERNIERS, 210, 211. LEVELS-mason's level, 21; spirit-level, 123; water-level, 18. MEASUREMENT OF AREA, 85-87. MEASUREMENT OF LENGTH-Foot-rule, 5; standard unit, Parallels, 56-60; carpenter's gauge, 59. POLYGONS-Geometrical rose, 216; ornamental windows, PROPORTION-Diagonal scale, 105; pantograph, 114; QUADRILATERALS—Isosceles trapezoids, 67, 68; rectangu- Right-angles and perpendiculars, 12, 16-19. re- Spirals and volutes (balustrades, Gothic windows, Ionic Tangents and arcs, 145-148. Transmission of motion by rollers, 171; by cogs, 172-174. 138, 156, 183, 197, 229, 257. Unit of length, 5; of area, So. Butler & Tanner, The Selwood Printing Works, Frome, and London. |