What Is a Pyramid?
©Lorena Loo
©Lorena Loo
Most people think a pyramid is a 3-dimensional object with a square base and triangular faces meeting at a single point (apex) like the pyramids in Egypt. Well that is only partly true for the pyramids in Egypt are special cases of the more general form of what constitutes a pyramid.
There is a very precise, concise way to define what a pyramid is (for the masochist or math-head, click on image to immediate right: Unfortunately, it is in a mathematical terminology that would leave most people scratching their heads and saying "Hunh?"
So let's explain what a pyramid is in plain, simpler language. It will require more words but we'll sacrifice conciseness for understandability here.
A pyramid is a 3-dimensional object with a base and triangular faces. Its base is a polygon, a word which literally means many angles like in polygamy (many marriages). Now the polygon is a closed figure meaning that any two lines forming adjacent sides must meet at a common point called a vertex. A polygon is also a 2-dimensional figure or object. Two-dimensional figures are called plane figures or planar figures. It is just a matter of getting used to the terminology.
For example, the figure to the left below is an open two dimensional (plane) figure and therefore not a polygon. The figure to the right is a closed two dimensional figure and hence a polygon. Many angles but also many sides and as many angles as there are sides.
There is a very precise, concise way to define what a pyramid is (for the masochist or math-head, click on image to immediate right: Unfortunately, it is in a mathematical terminology that would leave most people scratching their heads and saying "Hunh?"
So let's explain what a pyramid is in plain, simpler language. It will require more words but we'll sacrifice conciseness for understandability here.
A pyramid is a 3-dimensional object with a base and triangular faces. Its base is a polygon, a word which literally means many angles like in polygamy (many marriages). Now the polygon is a closed figure meaning that any two lines forming adjacent sides must meet at a common point called a vertex. A polygon is also a 2-dimensional figure or object. Two-dimensional figures are called plane figures or planar figures. It is just a matter of getting used to the terminology.
For example, the figure to the left below is an open two dimensional (plane) figure and therefore not a polygon. The figure to the right is a closed two dimensional figure and hence a polygon. Many angles but also many sides and as many angles as there are sides.
Now that you know what a polygon is and that a pyramid has a polygonal base and triangular faces, you know there are many different types of pyramids. Illustrated below are a few examples in a view from above. These are all pyramids with regular polygons as bases: equilateral triangle (all sides of the triangle are of equal length), square, regular pentagon, regular hexagon. Each of the triangular faces of the pyramids below are shaded a different color.
© Copyright Lorena Loo
For simplicity, let's only talk about pyramids with regular polygons as bases. We can form the line that joins the apex of the pyramid to the exact center of the base. If that line is at right angles (90º) to the base of the pyramid, then we have what is called a regular pyramid. The length of that joining line is the height or altitude of the regular pyramid. For a regular pyramid, the triangular faces of that pyramid are all identical. That is, corresponding lengths are all equal, corresponding angles are all equal.
Now let's look at another very special type of pyramid. When we increase the number of sides in a regular polygon, the polygon assumes more and more of a circular shape. A circle then is actually a special type of regular polygon with an infinite number of sides as illustrated below. We started with a regular pentagon, then increased the number of sides from 5 to 10, 15, 20 and finally 30 sides proceeding from left to right in the diagram. See how the polygon progresses toward more and more of a circular shape as its number of sides increase?
If we were to take a regular pyramid and increase the number of sides of its regular polygonal base, the base becomes increasingly more circular. At the same time, the number of its triangular faces increases so that they become "thinner" and "thinner," gradually melding into one smooth surface. When the polygonal base has an infinite number of sides, it becomes a circle and the triangular faces have melded completely into a smooth surface. Our regular pyramid has then become a cone. We can think of a cone then as a regular pyramid with an infinite number of sides to its polygonal base.
Like everything else in this world, shapes which include pyramids are defined and categorized. It helps in clarifying when you are talking about a pyramid, which type or kind of pyramid you are referring to. It's just like referring to your uncle by his name, Uncle John, instead of that tall uncle with brown hair, green eyes, a slight lisp and who tells lousy jokes.
But also, if geometric forms do have effects on a universal energy as researchers in the field have found, then the type of pyramid is important. A regular pyramid with a square base will not produce the same effect as a pyramid with an octagonal base or a tetrahedron. Or at the very least, will not be as effective.
But also, if geometric forms do have effects on a universal energy as researchers in the field have found, then the type of pyramid is important. A regular pyramid with a square base will not produce the same effect as a pyramid with an octagonal base or a tetrahedron. Or at the very least, will not be as effective.
Text and images (unless otherwise indicated in the credits) are copyrighted
© by Lorena Loo
In the instances where images in the public domain have been modified as in the case of geometrizing them, the modified images are no longer public domain but the copyright of the author who made the modifications. Here that means me, Lorena Loo. That is by copyright law.
© by Lorena Loo
In the instances where images in the public domain have been modified as in the case of geometrizing them, the modified images are no longer public domain but the copyright of the author who made the modifications. Here that means me, Lorena Loo. That is by copyright law.
This is an open 4-sided plane figure. It is not a polygon.
This is a closed 4-sided plane figure. It is a a polygon.
A triangle is a three-sided polygon, a rectangle a four-sided polygon, a pentagon a five-sided polygon, etc. When the sides of a polygon are all of equal length and its angles are all equal in size, then you have a regular polygon. A square, for example, is a regular polygon, a special type of rectangle where all four sides are equal in length and all four interior angles are all the same size (90º).
Polygons of sides 3, 4, 5 and 6 called convex polygons from left to right
These are also examples of polygons called concave polygons
These are regular polygons where the sides of the polygons are all of equal length and its angles are all the same size
Example of a right pyramid. Base is square. Red line is altitude or height of pyramid, the line joining its apex with the center of the base. The angle between that line and the base is 90º.
Some of you are familiar with a tetrahedron. But now that you know what a pyramid is, you might wonder then if a tetrahedron is not a pyramid. The answer is yes, indeed it is. A tetrahedron is a special type of pyramid with its base and three faces comprised of identically-sized equilateral triangles.
