Overview: What Is Solid Geometry?
Solid geometry was developed after plane geometry as a way to describe the three-dimensional world and the objects in it. In the ideal three-dimensional world, objects exist with faces and angles, depth and volume. Objects are regular, or consist of a combination of regular objects, just as two-dimensional figures can be made from a combination of other two dimensional figures. Solid geometry studies the properties, dimensions, and relationships of regular solids, as well as pyramids, cones, cylinders, prisms, spheres, and other solid objects.
What Are Regular Solids?
Regular solids have congruent faces, and the same number of faces meet at each vertex. A solid with four regular triangular faces is called a tetrahedron, a cube has six square faces, an octahedron has eight triangular faces, a dodecahedron has 12 faces that are shaped like regular pentagons, and an icosahedron has 20 triangular faces. They are symmetrical, and the only polygons that can fulfill all the requirements for sides, edges, and angles are triangles, squares, and pentagons.
Pyramids, Cones, Cylinders, Prisms
Some other solid figures do not fit the image of strict regularity. For example, polygons other than triangles form the base of pyramids, yet the other faces are triangular. Cones usually have a circular or elliptical base and an infinite number of faces tapering to a single point. Cylinders are formed by the solid enclosed when two circles in parallel planes are joined. Prisms are a special class of solids with a polygonal base, a translation of that base in a parallel plane (similar to a cylinder) and sides that join them.
Spheres and Spheroids
Spheres are perfectly round in three dimensions. They have many distinct properties, such as every point on the surface of the sphere is the same distance from the center, and that the width and the girth do not vary in a perfect sphere. Objects that are close to spheres are spheroids and approximate many of the properties of spheres. They may be elliptical rather than perfectly round, or polyhedrons that have a large number of sides. Spherical geometry is the branch of trigonometry dealing with spheres and spheroids.
Applications of Solid Geometry
Many geometric solids have applications in nature and technology. Common crystal shapes include the tetrahedron, cube, and octahedron. In addition, most viruses have the shape of a regular polyhedron because a basic unit protein can be repeated and packed into the smallest space possible. The most common sphere is a ball or globe, and the most common spheroid is the Earth itself.
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