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Study Notes 023, Solid Geometry 1:
By William E. Steinman:
The derivations of some of the equations we use in solid geometry are beyond the scope of this set of notes. They require knowledge of calculus so, to some extent, we just take them on faith. However, in many cases we can step through the logic to arrive at an intuitive understanding.
A polyhedron is a solid bounded by planes.
More completely, a polyhedral angle is formed when three or more planes meet at a common point.
That common point is called the vertex.
The lines where the planes intersect are called the edges.
The planes between the edges are called the faces.
Confusing enough?
Let’s draw a polyhedron.
Imagine that this sketch is a transparent three-dimensional solid figure.
Got it?
Okay!
A regular polyhedron has faces that are congruent regular polygons with equal polyhedral angles.
Remember a regular polygon is a polygon having equal sides like an equilateral triangle.
And, congruent means coinciding exactly when superimposed.
What we just said is a regular polyhedron is made up of regular polygons.
So the above figure is a sketch of a regular polyhedron.
An interesting example of a polyhedron is a prism.
A prism has two congruent polygon faces which are parallel. These are called the bases of the prism.
Planes through the bases form the other faces of the prism. These faces are called lateral faces of the prism.
Let’s draw a prism.
Again, try to imagine this as a transparent solid figure.
In this prism the lateral edges are perpendicular to the base. Hence, it is a right prism, in this case a right triangular prism.
A transparent body of this form, often of glass or plastic can be used for separating white light passed through it into a spectrum.
Prisms are valuable tools in many scientific pursuits.
Note that a prism need not be a triangular prism. It could be a hexagonal or square prism and still meet the required definition.
A regular prism has bases that are regular polygons.
The altitude of a prism is just the distance between its bases.
A parallelepiped is a prism whose bases are parallelograms.
Remember a parallelogram is a four-sided plane figure with opposite sides parallel.
A right parallelepiped is a parallelepiped in which all the faces are rectangles.
A cube is an example of a right parallelepiped.
A pyramid is a polyhedron in which one face is a polygon. This is called the base. The other lateral faces are then triangles meeting at a common vertex point.
A right pyramid has a base that is a regular polygon and an altitude that meets the base at its center.
It would look something like this imagining that this was a solid transparent figure.
Of course, it does not have to be a square regular pyramid to meet the specification. The base could be any regular polygon.
A cylinder is a solid figure bounded by two parallel planes and a surface generated by a line segment rotating parallel to itself.
Huh?
Okay, lets make it simpler.
Remember how we generated a circle by rotating a radius around a fixed point on a plane.
The surface of the cylinder is generated the same way by extending the end radius point into a line segment from one plane to another parallel plane.
The sections of the parallel planes are the bases.
In this example the fixed point or center is o, the radius is r and the end point is p. The line segment is h. When we rotate the line segment around o we generate the surface of the cylinder.
Notice that a cylinder need not be a right circular cylinder.
This example is clearly not.
A right circular cylinder is generated by rotating a rectangle around one of its sides. A beer can is a common example.
Consider the parallelogram above consisting of sides op, ox, py, and xy.
If that were a rectangle it would generate a right circular cylinder when we rotated it around the side ox.
Here is what it would look like.
The axis of a right circular cylinder is the line joining the center of its circular bases. The altitude is the length of its axis.
Enough for today!