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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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434:, 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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Tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the
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1376:. Vol. Book II. Stereometry. Translated by Givental, Alexander. Sumizdat.
1081:{\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1:}
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257:, 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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828:, a second base translated and rotated.sides]] of the two bases
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555:
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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
568:). This more restrictive type of cuboid is also known as a
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copy (rigidly moved without rotation) of the first, and
278:
Basic topics in solid geometry and stereometry include:
1242:, Britannica Educational Publishing, 2010, pp. 67–68.
37:"Solid object" redirects here. Not to be confused with
560:
Some sources also require that each of the faces is a
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1184:
algebra, which are important for higher dimensions.
924:A right circular cone and an oblique circular cone
74:. Unsourced material may be challenged and removed.
1280:
1080:
476:A hexahedron with three pairs of parallel faces
419:For a more complete list and organization, see
888:kite faces around an axis, with half offsets
8:
1107:ellipsoid (bottom right, a=4.5, b=6, c=3)]]
518:, except that its faces are not squares but
564:(so each pair of adjacent faces meets in a
394:of three dimensions (leading to a proof of
1061:
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1024:
1018:
1007:
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952:A right and an oblique circular cylinder
134:Learn how and when to remove this message
966:by deforming it by means of directional
440:
145:
1263:Paraphrased and taken in part from the
1251:
1232:
1287:. Cambridge University Press. p.
962:A surface that may be obtained from a
1279:Robertson, Stewart Alexander (1984).
7:
1315:Elements of Synthetic Solid Geometry
72:adding citations to reliable sources
678:mapping any vertex onto any other)
511:where all edges are the same length
473:), each of which is a parallelogram
25:
1312:Dupuis, Nathan Fellowes (1893).
1240:The Britannica Guide to Geometry
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1143:that is generated by rotating a
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59:needs additional citations for
205:Solid geometry deals with the
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213:of various solids, including
1266:1911 Encyclopædia Britannica
641:Small stellated dodecahedron
398:by using an extra dimension)
970:, or more generally, of an
773:, a second base which is a
421:List of mathematical shapes
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1102:(bottom left, a=b=5, c=3),
943:A solid elliptic cylinder
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554:is the same as that of a
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387:Advanced topics include:
183:of 3D space bounded by a
32:the film of the same name
1391:Euclidean solid geometry
897:tetragonal trapezohedron
860:center with two apexes.
30:Not to be confused with
18:Three-dimensional region
1372:Kiselev, A. P. (2008).
982:Examples of ellipsoids
589:right rectangular prism
483:of which the base is a
190:; for example, a solid
1283:Polytopes and Symmetry
1082:
154:
1318:. Macmillan. p.
1083:
972:affine transformation
628:and sharp corners or
415:List of solid figures
149:
1194:3D computer graphics
988:
869:triangular bipyramid
430:is the surface of a
408:descriptive geometry
68:improve this article
1340:Weisstein, Eric W.
791:corresponding sides
738:and a vertex point
674:(i.e., there is an
652:Toroidal polyhedron
443:
392:projective geometry
1206:Euclidean geometry
1147:around one of its
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659:Uniform polyhedron
570:rectangular cuboid
441:
396:Desargues' theorem
155:
1170:analytic geometry
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793:of the two bases
785:(necessarily all
712:Snub dodecahedron
672:vertex-transitive
544:convex polyhedron
170:three-dimensional
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16:(Redirected from
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185:two-dimensional
173:Euclidean space
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762:comprising an
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194:consists of a
188:closed surface
175:(3D space). A
158:Solid geometry
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85: –
84:
80:
79:Find sources:
73:
69:
63:
62:
57:This article
55:
51:
46:
45:
40:
33:
19:
1373:
1352:. Retrieved
1345:
1335:
1323:. Retrieved
1314:
1307:
1282:
1274:
1264:
1259:
1252:Kiselev 2008
1247:
1239:
1235:
1191:
1188:Applications
1167:
1104:
1097:
1096:
1089:
885:
853:
818:
778:
763:
728:
593:rectangular
592:
588:
582:rectangular
581:
577:
574:right cuboid
573:
569:
501:Rhombohedron
425:
386:
380:hyperboloids
339:dodecahedron
277:
251:Pythagoreans
248:
207:measurements
204:
177:solid figure
176:
161:
157:
156:
153:of one sheet
130:
121:
111:
104:
97:
90:
78:
66:Please help
61:verification
58:
39:Rigid object
1325:December 1,
1135:Hyperboloid
695:Tetrahedron
624:, straight
566:right angle
450:Definitions
343:icosahedron
321:tetrahedron
301:solid angle
223:polyhedrons
221:(and other
162:stereometry
151:Hyperboloid
1385:Categories
1366:References
1354:2019-11-04
1164:Techniques
882:polyhedron
850:polyhedron
815:polyhedron
789:) joining
775:translated
760:polyhedron
725:polyhedron
613:Polyhedron
584:hexahedron
471:hexahedron
467:polyhedron
426:Whereas a
376:paraboloid
335:octahedron
323:and other
259:Platonists
94:newspapers
1348:MathWorld
1145:hyperbola
1105:tri-axial
958:Ellipsoid
858:polygonal
844:Bipyramid
823:polygonal
809:Antiprism
768:polygonal
733:polygonal
692:(Regular)
619:polygonal
562:rectangle
403:polyhedra
372:ellipsoid
352:cylinders
284:incidence
231:cylinders
1374:Geometry
1346:Wolfram
1200:See also
1099:spheroid
968:scalings
930:Cylinder
676:isometry
630:vertices
436:cylinder
401:further
368:spheroid
364:quadrics
325:pyramids
215:pyramids
200:interior
198:and its
166:geometry
124:May 2014
1342:"Lemon"
1221:Surface
1141:surface
856:-sided
821:-sided
766:-sided
731:-sided
719:Pyramid
670:and is
453:Images
263:Eudoxus
245:History
211:volumes
179:is the
164:is the
108:scholar
1295:
1182:matrix
1174:vector
1091:sphere
964:sphere
914:vertex
884:with 2
781:other
709:Unform
536:Cuboid
520:rhombi
447:Figure
428:sphere
362:other
358:sphere
330:prisms
311:cuboid
288:planes
274:Topics
268:radius
219:prisms
196:sphere
181:region
110:
103:
96:
89:
81:
1227:Notes
1211:Shape
1114:Lemon
783:faces
754:Prism
668:faces
626:edges
622:faces
617:Flat
591:, or
481:prism
348:cones
292:lines
237:(and
235:cones
227:cubes
115:JSTOR
101:books
1327:2018
1293:ISBN
1180:and
1172:and
1120:lens
910:apex
904:Cone
826:base
771:base
736:base
699:Cube
697:and
556:cube
516:cube
432:ball
378:and
356:the
350:and
319:the
307:cube
305:the
299:and
290:and
249:The
192:ball
87:news
1196:.
912:or
666:as
438:.
286:of
241:).
225:),
209:of
168:of
160:or
70:by
1387::
1344:.
1320:53
1291:.
1289:75
1139:A
1118:A
880:A
848:A
813:A
758:A
723:A
587:,
580:,
576:,
572:,
542:A
514:A
507:A
479:A
465:A
374:,
370:,
366::
341:,
337:,
313:,
309:,
270:.
261:.
233:,
229:,
217:,
202:.
1357:.
1329:.
1301:.
1269:.
1254:.
1076::
1073:1
1070:=
1063:2
1059:c
1053:2
1049:z
1043:+
1036:2
1032:b
1026:2
1022:y
1016:+
1009:2
1005:a
999:2
995:x
886:n
854:n
819:n
779:n
764:n
729:n
598:.
423:.
410:.
382:.
137:)
131:(
126:)
122:(
112:·
105:·
98:·
91:·
64:.
41:.
34:.
20:)
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