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established their measurement, proving the pyramid and cone to have one-third the volume of a prism and cylinder on the same base and of the same height. He was probably also the discoverer of a proof that the volume enclosed by a sphere is proportional to the cube of its
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423:, for other solid figures it is sometimes ambiguous whether the term refers to the surface of the figure or the volume enclosed therein, notably for a
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1111:(or less than half of a circular arc) rotated about an axis passing through the endpoints of the lens (or arc)
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Tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the
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1365:. Vol. Book II. Stereometry. Translated by Givental, Alexander. Sumizdat.
1070:{\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1:}
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246:, but the pyramid, prism, cone and cylinder were not studied until the
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Various techniques and tools are used in solid geometry. Among them,
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Field of mathematics dealing with three-dimensional
Euclidean spaces
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techniques have a major impact by allowing the systematic use of
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Major types of shapes that either constitute or define a volume.
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Straight parallel sides and a circular or oval cross section
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A major application of solid geometry and stereometry is in
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557:). This more restrictive type of cuboid is also known as a
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copy (rigidly moved without rotation) of the first, and
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Basic topics in solid geometry and stereometry include:
1231:, Britannica Educational Publishing, 2010, pp. 67–68.
26:"Solid object" redirects here. Not to be confused with
549:
Some sources also require that each of the faces is a
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algebra, which are important for higher dimensions.
913:A right circular cone and an oblique circular cone
63:. Unsourced material may be challenged and removed.
1269:
1069:
1371:
465:A hexahedron with three pairs of parallel faces
408:For a more complete list and organization, see
1328:
877:kite faces around an axis, with half offsets
1096:ellipsoid (bottom right, a=4.5, b=6, c=3)]]
507:, except that its faces are not squares but
553:(so each pair of adjacent faces meets in a
383:of three dimensions (leading to a proof of
1267:
941:A right and an oblique circular cylinder
123:Learn how and when to remove this message
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955:by deforming it by means of directional
403:
134:
1360:
1252:Paraphrased and taken in part from the
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1300:
1276:. Cambridge University Press. p.
951:A surface that may be obtained from a
1268:Robertson, Stewart Alexander (1984).
1304:Elements of Synthetic Solid Geometry
61:adding citations to reliable sources
32:
667:mapping any vertex onto any other)
500:where all edges are the same length
462:), each of which is a parallelogram
13:
14:
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1301:Dupuis, Nathan Fellowes (1893).
1229:The Britannica Guide to Geometry
1142:
1132:that is generated by rotating a
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194:Solid geometry deals with the
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202:of various solids, including
1255:1911 Encyclopædia Britannica
630:Small stellated dodecahedron
387:by using an extra dimension)
7:
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959:, or more generally, of an
762:, a second base which is a
410:List of mathematical shapes
10:
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1091:(bottom left, a=b=5, c=3),
932:A solid elliptic cylinder
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543:is the same as that of a
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376:Advanced topics include:
262:
172:of 3D space bounded by a
21:the film of the same name
1380:Euclidean solid geometry
1215:
886:tetragonal trapezohedron
849:center with two apexes.
19:Not to be confused with
1361:Kiselev, A. P. (2008).
971:Examples of ellipsoids
578:right rectangular prism
472:of which the base is a
179:; for example, a solid
1272:Polytopes and Symmetry
1071:
143:
1307:. Macmillan. p.
1072:
961:affine transformation
617:and sharp corners or
404:List of solid figures
138:
1183:3D computer graphics
977:
858:triangular bipyramid
419:is the surface of a
397:descriptive geometry
57:improve this article
1329:Weisstein, Eric W.
780:corresponding sides
727:and a vertex point
663:(i.e., there is an
641:Toroidal polyhedron
432:
381:projective geometry
1195:Euclidean geometry
1136:around one of its
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648:Uniform polyhedron
559:rectangular cuboid
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385:Desargues' theorem
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1159:analytic geometry
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782:of the two bases
774:(necessarily all
701:Snub dodecahedron
661:vertex-transitive
533:convex polyhedron
159:three-dimensional
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458:with six faces (
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72:"Solid geometry"
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791:hexagonal prism
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535:bounded by six
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242:dealt with the
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162:Euclidean space
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751:comprising an
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177:closed surface
164:(3D space). A
147:Solid geometry
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865:Trapezohedron
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74: –
73:
69:
68:Find sources:
62:
58:
52:
51:
46:This article
44:
40:
35:
34:
29:
22:
1362:
1341:. Retrieved
1334:
1324:
1312:. Retrieved
1303:
1296:
1271:
1263:
1253:
1248:
1241:Kiselev 2008
1236:
1228:
1224:
1180:
1177:Applications
1156:
1093:
1086:
1085:
1078:
874:
842:
807:
767:
752:
717:
582:rectangular
581:
577:
571:rectangular
570:
566:
563:right cuboid
562:
558:
490:Rhombohedron
414:
375:
369:hyperboloids
328:dodecahedron
266:
240:Pythagoreans
237:
196:measurements
193:
166:solid figure
165:
150:
146:
145:
142:of one sheet
119:
110:
100:
93:
86:
79:
67:
55:Please help
50:verification
47:
28:Rigid object
1314:December 1,
1124:Hyperboloid
684:Tetrahedron
613:, straight
555:right angle
439:Definitions
332:icosahedron
310:tetrahedron
290:solid angle
212:polyhedrons
210:(and other
151:stereometry
140:Hyperboloid
1374:Categories
1355:References
1343:2019-11-04
1153:Techniques
871:polyhedron
839:polyhedron
804:polyhedron
778:) joining
764:translated
749:polyhedron
714:polyhedron
602:Polyhedron
573:hexahedron
460:hexahedron
456:polyhedron
415:Whereas a
365:paraboloid
324:octahedron
312:and other
248:Platonists
83:newspapers
1337:MathWorld
1134:hyperbola
1094:tri-axial
947:Ellipsoid
847:polygonal
833:Bipyramid
812:polygonal
798:Antiprism
757:polygonal
722:polygonal
681:(Regular)
608:polygonal
551:rectangle
392:polyhedra
361:ellipsoid
341:cylinders
273:incidence
220:cylinders
1363:Geometry
1335:Wolfram
1189:See also
1088:spheroid
957:scalings
919:Cylinder
665:isometry
619:vertices
425:cylinder
390:further
357:spheroid
353:quadrics
314:pyramids
204:pyramids
189:interior
187:and its
155:geometry
113:May 2014
1331:"Lemon"
1210:Surface
1130:surface
845:-sided
810:-sided
755:-sided
720:-sided
708:Pyramid
659:and is
442:Images
252:Eudoxus
234:History
200:volumes
168:is the
153:is the
97:scholar
1284:
1171:matrix
1163:vector
1080:sphere
953:sphere
903:vertex
873:with 2
770:other
698:Unform
525:Cuboid
509:rhombi
436:Figure
417:sphere
351:other
347:sphere
319:prisms
300:cuboid
277:planes
263:Topics
257:radius
208:prisms
185:sphere
170:region
99:
92:
85:
78:
70:
1216:Notes
1200:Shape
1103:Lemon
772:faces
743:Prism
657:faces
615:edges
611:faces
606:Flat
580:, or
470:prism
337:cones
281:lines
226:(and
224:cones
216:cubes
104:JSTOR
90:books
1316:2018
1282:ISBN
1169:and
1161:and
1109:lens
899:apex
893:Cone
815:base
760:base
725:base
688:Cube
686:and
545:cube
505:cube
421:ball
367:and
345:the
339:and
308:the
296:cube
294:the
288:and
279:and
238:The
181:ball
76:news
1185:.
901:or
655:as
427:.
275:of
230:).
214:),
198:of
157:of
149:or
59:by
1376::
1333:.
1309:53
1280:.
1278:75
1128:A
1107:A
869:A
837:A
802:A
747:A
712:A
576:,
569:,
565:,
561:,
531:A
503:A
496:A
468:A
454:A
363:,
359:,
355::
330:,
326:,
302:,
298:,
259:.
250:.
222:,
218:,
206:,
191:.
1346:.
1318:.
1290:.
1258:.
1243:.
1065::
1062:1
1059:=
1052:2
1048:c
1042:2
1038:z
1032:+
1025:2
1021:b
1015:2
1011:y
1005:+
998:2
994:a
988:2
984:x
875:n
843:n
808:n
768:n
753:n
718:n
587:.
412:.
399:.
371:.
126:)
120:(
115:)
111:(
101:·
94:·
87:·
80:·
53:.
30:.
23:.
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