Saturday, May 2, 2015

Unizor - Geometry3D - Elements - Pyramids

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Consider a plane α that we will call a base plane and a polygon ABCDEF on this plane (we specify 6-sided polygon, but it's not important, any polygon will do). This polygon will be used as a directrix of a conical surface we will construct.
Also assume there is a point S outside this plane that we will use as an apex of our conical surface.

Let's construct a conical surface σ using polygon ABCDEF as a directrix and point S as an apex.

Pyramid is an object in solid geometry formed by a part of a conical surface σ between a base plane α and an apex S and a part of the base plane inside that conical surface - polygon ABCDEF on the base plane α.

Points A, B, C, D and all others are called vertices of a pyramid. Apex S is also a vertex, though a special one.
Segments AB, BC, DS and all others are called edges of a pyramid.
Polygon ABCDEF is called a base of a pyramid.
Triangles ABS, BCS and others are called sides of a pyramid.
Base and sides of a pyramid are generically referred to as faces. Sides are also referred to as lateral faces.
A perpendicular from an apex to a base of a pyramid is its altitude or height.

Another classification of pyramids is by their bases.
If a base is a triangle, the pyramid is called triangular pyramid.
If a base is a rectangle, the pyramid is called rectangular pyramid.
If a base is a square, the pyramid is called square pyramid (Egyptian pyramids are of this kind).
If a base is a hexagon, the pyramid is called hexagonal pyramid.
If a base is a regular N-sided polygon and a perpendicular from the apex onto the base falls in a center of this regular N-sided polygon, the pyramid might be called N-pyramid.

Now let's imagine that we have another plane β parallel to base α and intersecting a pyramid's altitude somewhere between an apex and a base.
A geometric object consisting of a conical surface between planes α and β and parts of these planes lying inside the conical surface - polygon ABCDEF on the base plane α (we can call it "bottom") and a corresponding polygon A'B'C'D'E'F' on the plane β (we can call it "top") - is called a truncated pyramid (like the one on the 1 dollar bill of the US currency - one of the Masonic symbols).
All the terminology of pyramids is the same for truncated pyramids. The altitude of a truncated pyramid is a distance between "top" and "bottom" bases along a common perpendicular.

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