San José State University |
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Theorems Concerning Areas and Volumes That Includes Circles and Spheres as Limits |
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Consider a symmetric polygon of n sides. Its area is found by decomposing it into n equilateral triangles of height h and base length b. Its area A is then given by
The distance h is the minimum distance from the center of the polygon to its boundary edge. Thus the theorem is that for a symmetric polygon its area A is equal to ½hB, where B is the length of its boundary and h is the minimum distance from the center of the polygon to its boundary edge.
For a circle h is its radius r and B is its circumference 2πr. Thus
A symmetric polyhedron of n sides can be decomposed into n pyramids of height h and base area b. Each of these has volume ⅓hb. The volume V of the polyhedron is then given by
The area B of the surface of the polyhedron is nb. The quantity h is again the minimum distance from the center of the polyhedron to its boundary surface. The theorem is thus that the volume V of a symmetric polyhedron is
For a sphere, h is its radius r and it bounary surface is 4πr². Thus
Now consider a line segment of length L. Its boundary is the two end points so B=2. The value h for it is L/2. So an analogous formula applies to the one dimensional case; i.e.,
Let d be the dimensionality of the symmetric polytope; 1 for line segments, 2 for polygons and 3 for polyhedra. The formula that applies to the three cases is
where W is the d-dimensional volume and U is the (d−1)-dimensional boundary surface.
Consider a Star-of-David. The area of the central hexagon is as above ½h_{1}(6b). The area of the points is 6(h_{2}b). Thus the total area is ½(h_{1}+h_{2})(6b)=½hB, where B is the border length of the central hexagon and h=h_{1}+h_{2}) is in the nature of a maximum distance from the center to outer border of the figure.
Likewise the volume of a pointed polyhedron is
with B being the area of the central polyhedron.
A star-type of a line segment is just a longer line segment so the extension of the theorem is trivial.
Symmetric figures can also be created by by sucessively rotating a polygon about an axis of symmetry by one n-th of a full circle. This creates a sheaf of polygons. Similarly a symmetric sheaf of line segments can be created.
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