To find the Area of a Segment of a Circle. Rule-Divide the height, or versed sine, by the diameter of the circle, and find the quotient in the column of versed sines. Then take out the corresponding area, in the next column on the right hand, and multiply it by the square of the diameter; this will give the area of the segment. Example.-Required the area of a segment of a circle, whose height is 3 feet, and the diameter of the circle 50 feet. 31=325; and 3 25÷50065. 065, as per table,=021659; and '021659 × 502=54 147500, the area required. Approximating Rule to find the Area of a Segment of a Circle. Rule.-Multiply the chord of the segment by the versed sine, divide the product by 3, and multiply the remainder by 2. Cube the height, or versed sine, find how often twice the length of the chord is contained in it, and add the quotient to the former product; this will give the area of the segment, very nearly. Example-Required the area of the segment of a circle, the chord being 12, and the versed sine 2. 12×2=24; 24÷3-8; and 8×2=16 23÷24=3333. Hence 16+3333=16.3333, the area of the segment, very nearly TABLE OF THE AREAS OF THE ZONES OF A CIRCLE. Versed Area of Versed Sine. Segment. Sine. 001 00100 044 00200200 045 003 00300046 004 00400 047 005 00500 048 006 00600049 007 00700 050 00800800 051 009 00900 052 010 01000 053 011 01100 054 17882 184 17975 185 18068 18161 012 01199 055 05489 098-09736 .141 13910 187 18254 016 01599 059 05886 102 10128 .14514294188 18347 017 01699 060 0598510310226.146 14389 189 18439 19018532 14581 191 18624 18717 06084 10410324 147 14485 06283106 10520-149 018 01799 061 019 01899 062 02001999 063 021 02099 064 '022 02199 065 06481 023 02299 066 06580 024 02399 067 06679 025 02499 068 06779 026 02598 069 06878 027 02698 070 028 02798 071 029 02898 072 030 02998073 031 03098 074 032 03197 075 18901 14676 192 14771 193 18809 108 10715151.14867194 10910813 152 14962195 18993 110 10910 153 1505719619085 111 11008 154 15153 197 19177 -112 11105 155 06977 113 11203156 07076114 11300157 0717511511397 158 07274 116 11495 159 07372 117 11592 07471 11811689 0757011911786 15248 198 19269 15343 199 19361 15438 200 19453 1553320119544 15627 20219636 160 15722 || 203 | 19727 16115817 204 19819 162 15911 163 033 03297 076 036 03596 079 037 03696 080 034 03397 077 07669 120 11883 205 19910 16006 206 20001 16101 207 20092 327 21620908 217 265 25201 314 29192 363 32793 412 35882 20998 266 25285 315 218 21088 26725370 316 219 21178 268 25454 317 220 21268 269 25539 318 221 21357 270 25623 319 222 21447 271 25707 320 22321536 272 25791 224 21626 273 25875 225 21715 274 25959 226 21805 275 26042 227 21894 276 26126 22821983 277 26209 229 22072 278 26292 29270 ·364 32862 413 35939 29347 || 365 32931 414 35995 36107 36162 29656|| 369 33202 418 36217 321 29733 370 33270|| ·419 36272 36326 36380 385 337 30937 386 34317 435 338 31011 387 34380 436 37153 339 31085 388 34443 437 37201 291 27361 340 31158 389 34506|| 438 37250 29227442 || 341 31231 390 34569 439 37298 31305 391 34631 440 37346 34253 || 434 37054 37104 29527685 344 31450 393 34756 442 37440 29627766 345 31523 394 34817 443 297-27846 346 31595 395 34879 444 249 23828 298 27927 347 31667 396 34940 445 37578 250 23915 299 .28007 348 31739 397 35001 446 37624 251 24001 30028087 349 31811 398 35061 447 37669 252 24088 301 28167 350 253 24174 30228247 351 25424260303-28326 352 255 24346 304 28406 353 256 24432 305 28485 354 257 24518 306 28564 355 32237 404 308 28722 357 32377 406 31229036 361 .462 38293470·38583 478 46338331 471 38617479 461 38255 469 38549 477 38808 485 39026 493 39120 38837 486 39050 494 39208 38866 487 39073 495 39222 38650 480 38895 488 39095 496 39236 38683 481 38922 489 39116 497 39248 38715 482 38949 49039137 498 38746 483 38975 491 39156 38777 484 39001 49239174 464 38369 472 465 38406473 466 38442 474 46738478 475 468 38514476 39258 499 39265 500 39269 USE OF THE ABOVE TABLE. To find the Area of a Circular Zone. Rule 1.-When the zone is less than a semicircle, divide the height by the longest chord, and seek the quotient in the column of versed sines. Take out the corresponding area, in the next column on the right hand, and multiply it by the square of the longest chord; the product will be the area of the zone. Example.-Required the area of a zone, whose longest chord is 50, and height 15. 15÷50=300; and 300, as per table,=28087. Hence, 28087 × 502=702·19, the area of the zone. Rule 2.-When the zone is greater than a semicircle, take the height on each side of the diameter of the circle, and find, by Rule 1, their respective areas; the areas of these two portions, added together, will be the area of the zone. Example.-Required the area of a zone, the diameter of the circle being 50, and the height of the zone on each side of the line which passes through the diameter of the circle 20 and 15, respectively. 20÷50=400; 400, as per table,=35182; and 35182 × 502= 879.56. 15÷50=300; 300, as per table,=28087; and 28087 × 502= 702.19. Hence, 879 56+702-19=1581-75. Length TABLE Of the Proportions of the Lengths of Circular Arcs. Height of Arc. of Arc. of Arc. of Arc. 144 1.0544 100 1.0265 Length 232 1.1379 276 1.1921 233 1·1390 277 1-1934 282 1.2001 283 12015 284 1.2028 285 1.2042 286 1.2056 287 1.2070 288 1.2083 289 1.2097 290 1.2120 291 1-2124 292 1-2138 293 1-2152 294 1.2166 207 11006|| 251 1 1603 295 1-2179 188 1.0917 101 10270 145 1.0552 189 1.0927 102 10275 146 10559|| 190 1·0936 234 1.1402 278 1.1948 103 10281 147 10567|| 191 | 1·0946 235 1.1414279 1.1961 104 10286 148 10574192 10956236 1.1425 230 1.1974 105 1.0291 149 10582 193 1.0965 237 1-1436 281 1.1989 106 1.0297 150 10590 194 1.0975 238 1.1448 107 1.0303 151 10597 195 10985 239 1·1460 108 1.0308 152 10605 196 10995 240 1·1471 109 10314 153 1.0613 197 11005241 1.1483 110 10320154 1.0621 198 11015 242 1·1495 111 10325 155 1.0629 199 11025|| 243 1.1507 112 1.0331 156 1.0637 200 11035 244 1·1519 113 10337 157 10645 201 1 1045 245 1.1531 114 10343 158 10653|| 202 11055|| 246 1.1543 115 1.0349 159 1.0661 203 11065 247 1.1555 116 10355 160 1.0669 204 11075|| 248 1·1567 117 1.0361 161 10678 205 1 1085 249 1.1579 118 10367 162 10686|| 206 11096 250 1.1591 119 10373 163 10694 120 1.0380 164 10703 208 121 1.0386 165 10711 209 11127 253 1.1628 122 1.0392 166 10719 210 1'1137|| 254 1·1640 123 1.0399 167 1.0728 211 11148 255 1.1653 124 1.0405 168 10737 212 1'1158 256 1.1665 300 1.2250 125 1.0412 169 10745 213 1'1169 257 1.1677 301 1.2264 126 1.0418 170 10754 214 1'1180 258 1.1690 302 1.2278 127 1.0425 171 1'0762 215 1'1190 259 1·1702 303 1-2292 128 1.0431 •172 1'0771 216 1 1201 260 1.1715 304 1.2306 129 1.0438 173 10780 217 11212 261 130 1.0445 174 1'0789 218 131 1.0452175 10798 219 132 1.0458 || 176 10807 220 1 1245 || 264 133 1.0465177 10816 221 1.1256|| 265 1.1778 309 1.2378 134 1.0472178 1.0825 222 1.1266 266 1.1791 310 1.2393 135 1.0479 || 179 1.0834 || 223 1.1277267 1.1804 | 311 1·2407 136 1.0486180 1.0843 224 1.1289 268 1.1816 312 1.2422 137 1.0493181 1.0852 225 1.1300 269 1.1829 313 1.2436 138 1.0500 182 1.0861 226 1.1311 270 1.1843 314 1.2451 139 1.0508183 1.0870 227 11322 || 271 1.1856 315 1.2465 140 1.0515 184 1.0880228 1.1333 272 1.1869 316 1.2480 141 1.0522185 1.0889229 1.1344 273 1.1882 317 1.2495 142 1.0529186 1.0898 230 1.1356|| 274 1.1897 318 1.2510 143 1.0537187 1.0908231 11367 275 1.1908 319 1.2524 |