457:
29:
37:
56:
with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow rectangle-section ring is produced. If the revolved figure is a
285:
235:
The volume (V) and surface area (S) of a toroid are given by the following equations, where r is the radius of the circular section, and R is the radius of the overall shape.
328:
153:
The volume (V) and surface area (S) of a toroid are given by the following equations, where A is the area of the square section of side, and R is the radius of revolution.
225:
189:
137:
measured from the center of the section rotated. For symmetrical sections volume and surface of the body may be computed (with circumference
403:
Stewart, B.; "Adventures Among the
Toroids:A Study of Orientable Polyhedra with Regular Faces", 2nd Edition, Stewart (1980).
339:
241:
484:
291:
20:
195:
159:
89:
349:
53:
100:
8:
92:
73:
479:
434:
413:
Carroll, Gregory T.; Jongejan, Mahthild G. M.; Pijper, Dirk; Feringa, Ben L. (2010).
380:
76:. In this context a toroid need not be circular and may have any number of holes. A
426:
461:
344:
473:
438:
383:
415:"Spontaneous generation and patterning of chiral polymeric surface toroids"
119:
456:
430:
414:
388:
122:
created by rotating a disk, and should not be confused with toroids.
354:
125:
Toroidal structures occur in both natural and synthetic materials.
115:
28:
114:
The torus is an example of a toroid, which is the surface of a
58:
359:
85:
62:
41:
412:
36:
294:
244:
198:
162:
322:
279:
219:
183:
133:A toroid is specified by the radius of revolution
471:
378:
16:Surface of revolution with a hole in the middle
84:can be seen as approximating the surface of a
35:
27:
472:
379:
340:Toroidal inductors and transformers
13:
230:
14:
496:
449:
280:{\displaystyle V=2\pi ^{2}r^{2}R}
455:
148:
118:. Doughnuts are an example of a
406:
397:
372:
61:, then the object is called a
1:
460:The dictionary definition of
323:{\displaystyle S=4\pi ^{2}rR}
128:
7:
333:
72:is also used to describe a
10:
501:
18:
220:{\displaystyle S=2\pi RC}
184:{\displaystyle V=2\pi RA}
365:
32:A toroid using a square.
99:, of 1 or greater. The
21:Toroid (disambiguation)
324:
281:
221:
185:
45:
33:
325:
282:
222:
186:
54:surface of revolution
39:
31:
292:
242:
196:
160:
107:holed toroid is 2(1-
101:Euler characteristic
44:is a type of toroid.
19:For other uses, see
345:Toroidal propellers
74:toroidal polyhedron
431:10.1039/c0sc00159g
381:Weisstein, Eric W.
320:
277:
217:
181:
48:In mathematics, a
46:
34:
145:of the section):
492:
485:Geometric shapes
459:
443:
442:
419:Chemical Science
410:
404:
401:
395:
394:
393:
376:
329:
327:
326:
321:
313:
312:
286:
284:
283:
278:
273:
272:
263:
262:
226:
224:
223:
218:
190:
188:
187:
182:
500:
499:
495:
494:
493:
491:
490:
489:
470:
469:
452:
447:
446:
411:
407:
402:
398:
377:
373:
368:
336:
308:
304:
293:
290:
289:
268:
264:
258:
254:
243:
240:
239:
233:
231:Circular toroid
197:
194:
193:
161:
158:
157:
151:
131:
24:
17:
12:
11:
5:
498:
488:
487:
482:
468:
467:
451:
450:External links
448:
445:
444:
405:
396:
370:
369:
367:
364:
363:
362:
357:
352:
347:
342:
335:
332:
331:
330:
319:
316:
311:
307:
303:
300:
297:
287:
276:
271:
267:
261:
257:
253:
250:
247:
232:
229:
228:
227:
216:
213:
210:
207:
204:
201:
191:
180:
177:
174:
171:
168:
165:
150:
147:
130:
127:
15:
9:
6:
4:
3:
2:
497:
486:
483:
481:
478:
477:
475:
466:at Wiktionary
465:
464:
458:
454:
453:
440:
436:
432:
428:
424:
420:
416:
409:
400:
391:
390:
385:
382:
375:
371:
361:
358:
356:
353:
351:
348:
346:
343:
341:
338:
337:
317:
314:
309:
305:
301:
298:
295:
288:
274:
269:
265:
259:
255:
251:
248:
245:
238:
237:
236:
214:
211:
208:
205:
202:
199:
192:
178:
175:
172:
169:
166:
163:
156:
155:
154:
149:Square toroid
146:
144:
140:
136:
126:
123:
121:
117:
112:
110:
106:
102:
98:
94:
91:
87:
83:
79:
75:
71:
66:
64:
60:
55:
51:
43:
38:
30:
26:
22:
462:
422:
418:
408:
399:
387:
374:
234:
152:
142:
138:
134:
132:
124:
113:
108:
104:
103:χ of a
96:
81:
77:
69:
67:
49:
47:
25:
120:solid torus
90:topological
474:Categories
425:(4): 469.
439:2041-6520
389:MathWorld
306:π
256:π
209:π
173:π
141:and area
129:Equations
88:having a
68:The term
480:Topology
384:"Toroid"
355:Solenoid
334:See also
116:doughnut
350:Annulus
80:-holed
463:toroid
437:
82:toroid
70:toroid
59:circle
50:toroid
366:Notes
360:Helix
93:genus
86:torus
63:torus
52:is a
42:torus
435:ISSN
427:doi
111:).
476::
433:.
421:.
417:.
386:.
95:,
65:.
40:A
441:.
429::
423:1
392:.
318:R
315:r
310:2
302:4
299:=
296:S
275:R
270:2
266:r
260:2
252:2
249:=
246:V
215:C
212:R
206:2
203:=
200:S
179:A
176:R
170:2
167:=
164:V
143:A
139:C
135:R
109:g
105:g
97:g
78:g
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
