Finding the Regular Mean in Geometry
By Kathleen Cantor, 18 Jul 2020
Mathematics is like any other language. To work in it, you need to first understand a particular set of definitions and rules. That's why when faced with a phrase like "find the regular mean in geometry," step back and look at the language. In this case, we have three different words to interpret: regular, mean, and geometry.
Regular refers to a regular polygon with a unique set of attributes, not the typical definition of "normal." Whenever you see "regular" in mathematics, it is about a polygon.
- All sides are equal in length
- All angles are equal in degree
Typically, people think of squares. They are not wrong, but there are other polygons than can be regular: triangles, pentagons, and hexagons, for example. There are also regular polyhedrons. These are three-dimensional polygons such as cubes, tetrahedrons, and octahedrons. A polyhedron is regular when all its faces are regular polygons.
Mean is another way of writing "average." When paired with geometry, it clues us in that we are working in geometric dimensions as opposed to arithmetic. You will be finding the geometric mean.
For this particular topic, we will learn how to find the geometric mean of an irregular polygon and an irregular polyhedron. This way, you will better understand what makes a regular polygon and how to form one from an irregular polygon using geometric mean.
When and Why Use Geometric Mean
Unless specifically asked on a test, quiz, or by a teacher, different real-life situations require you to use the geometric mean. When it comes to mean, however, there are two different options: arithmetic and geometric. Understanding when to use the arithmetic mean can significantly help you figure out when to use geometric mean.
When finding the arithmetic mean, we are finding the average on a number line. This number line is generated with a low number, high number, and frequently, other values that fall in between. These numbers are added up and divided by the total amount of numbers on the line.
The formula for arithmetic mean is (A+B)/2.
The underlying question we are asking ourselves when finding the arithmetic mean is what number would each value (A and B) have to be to add up to the original total of A+B?
To prove our question, we take the mean and add it twice to make our original total before division.
The geometric mean is used to calculate growth. We would use geometric mean not when something is added to find a total, but rather multiplied to find a product.
The standard formula for geometric mean is √(AB).
In other words, we are finding the square root when provided with two values, the cubed root when dealing with three values, etc. We can also apply the same question used for the arithmetic mean but replacing addition terms with multiplication terms.
Geometric mean is commonly used to calculate investment growth. Say you have investments that grow by three different values over three consecutive years. Your investments changed according to different interest rates (multiplication). You now want to replicate that exact pattern of growth. You'll have to find out what the interest rate would have to be every year to recreate this pattern using the geometric mean.
We won't be going into investment calculations, though, so let's jump back into geometry.
Geometric Mean of Polygons & Polyhedrons
Remember, our base formula for geometric mean is √(AB). It may help to draw out these problems on a piece of paper to visualize it better.
Geometric Mean of Polygons
A rectangle is an irregular polygon -- its lengths are different. You want to figure out what the lengths of a rectangle's sides need to be to generate the same area as its regular square counterpart.
This question can be presented to you as: What is the geometric mean of 4 and 16?
On your rectangle, label Length A (height) = 4. Label Length B (width) = 16.
Plug these values into the formula.
Solve the multiplication within the formula. You can also think of this as finding the area of your rectangle.
Find the square root of 64.
Your geometric mean is 8! This means that to change your rectangle to a square in which the area remains the same, all four sides must be equal to 8. Let's prove this. Find the area of a square where lengths A and B are equal to 8.
The value 8, when multiplied twice, results in the original product of the formula.
Geometric Mean of Polyhedrons
Finding the geometric mean of a polyhedron involves adding a third number to the mix. Instead of dealing with the square root, we will be finding the cubed root.
The formula for geometric mean involving three numbers is 3√(ABC).
Let's start with an irregular polyhedron: a box. A box is a rectangle with an added dimension. Do you remember the formula for finding the volume (area in three dimensions)? It is length x width x height. You'll need this to properly label the box.
What is the geometric mean of 1, 7, and 49?
On your rectangle, label Length A (length) = 1. Label Length B (width) = 7. Label Length C (height) = 49.
Input these numbers in your formula.
Solve for volume.
Now, find the cubed root of 343.
To prove this, let's see if our irregular polyhedron's geometric mean will give us a regular polyhedron (a cube).
When put into terms of investment calculations, finding the geometric mean is a little daunting. It's better to practice with polygons and polyhedrons first and then advance to the larger concepts. Geometric means are nothing to be afraid of, though! Nothing in math should be feared.
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