Solid geometry

 ( Solid Geometry ) 

Solid geometry or stereometry is the geometry of three-dimensional Euclidean space (3D space). The Britannica Guide to Geometry, Britannica Educational Publishing, 2010, pp. 67–68. A solid figure is the region of 3D space bounded by a two-dimensional closed surface; for example, a solid ball consists of a sphere and its interior.

Solid geometry deals with the of of various solids, including pyramids, prisms, cubes (and other polyhedrons), cylinders, cones (including Frustum) and other solids of revolution..

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History The Pythagoreanism dealt with the , but the pyramid, prism, cone and cylinder were not studied until the Platonism. Eudoxus 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 radius.Paraphrased and taken in part from the 1911 Encyclopædia Britannica.

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Topics Basic topics in solid geometry and stereometry include:

• incidence of planes and lines
• dihedral angle and solid angle
• the cube, cuboid, parallelepiped
• the tetrahedron and other pyramids
• prisms
• octahedron, dodecahedron, icosahedron
• cones and cylinders
• the sphere
• other : spheroid, ellipsoid, paraboloid and .

Advanced topics include:

• projective geometry of three dimensions (leading to a proof of Desargues' theorem by using an extra dimension)
• further polyhedra
• descriptive geometry.

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List of solid figures Whereas a sphere is the surface of a ball, 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 cylinder.

+ Major types of shapes that either constitute or define a volume. ! Figure !! Definitions !! colspan=2	Images
A polyhedron with six faces (hexahedron), each of which is a parallelogram A hexahedron with three pairs of parallel faces A prism of which the base is a parallelogram |colspan=2
A parallelepiped where all edges are the same length A cube, except that its faces are not squares but rhombus |colspan=2
A convex polyhedron bounded by six quadrilateral faces, whose polyhedral graph is the same as that of a cube Stewart Alexander Robertson (1984). 9780521277396, Cambridge University Press. . ISBN 9780521277396 Some sources also require that each of the faces is a rectangle (so each pair of adjacent faces meets in a right angle). This more restrictive type of cuboid is also known as a rectangular cuboid, right cuboid, rectangular box, rectangular hexahedron, right rectangular prism, or rectangular parallelepiped. |colspan=2
Polyhedron	Flat faces, straight edges and sharp corners or vertices	Small stellated dodecahedron	Toroidal polyhedron
Uniform polyhedron	as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other)	(Regular) Tetrahedron and Cube	Uniform Snub dodecahedron
A polyhedron comprising an n-sided base and a vertex point	square pyramid
A polyhedron comprising an n-sided base, a second base which is a translated copy (rigidly moved without rotation) of the first, and n other faces (necessarily all ) joining corresponding sides of the two bases	hexagonal prism
A polyhedron comprising an n-sided base, a second base translated and rotated.sides]] of the two bases	square antiprism
A polyhedron comprising an n-sided center with two apexes.	triangular bipyramid
A polyhedron with 2 n kite faces around an axis, with half offsets	tetragonal trapezohedron
Cone	Tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex	A right circular cone and an oblique circular cone
Cylinder	Straight parallel sides and a circular or oval cross section	A solid elliptic cylinder	A right and an oblique circular cylinder
Ellipsoid	A surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation	Examples of ellipsoids	$\{x^2\; \backslash over\; a^2\}+\{y^2\; \backslash over\; b^2\}+\{z^2\; \backslash over\; c^2\}=1:$ sphere (top, a=b=c=4), spheroid (bottom left, a=b=5, c=3), tri-axial ellipsoid (bottom right, a=4.5, b=6, c=3)]]
Lemon	A lens (or less than half of a circular arc) rotated about an axis passing through the endpoints of the lens (or arc)
Hyperboloid	A surface that is generated by rotating a hyperbola around one of its principal axes

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Techniques Various techniques and tools are used in solid geometry. Among them, analytic geometry and vector techniques have a major impact by allowing the systematic use of linear equations and matrix algebra, which are important for higher dimensions.

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Applications A major application of solid geometry and stereometry is in 3D computer graphics.

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See also

• Euclidean geometry
• Shape
• Solid modeling
• Surface

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Notes

• Robert Baldwin Hayward (1890) The Elements of Solid Geometry via Internet Archive

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