Fundamentals of Geometry

Though it can seem complicated at times, geometry can essentially boil down to a few fundamental concepts. These concepts have been known for thousands of years, with their origins in several different ancient cultures. The ancient Greek mathematician Euclid is often considered the 'father' of geometry, with his fundamentals of Euclidean geometry serving as a basis for much of 'modern mathematics' understanding of geometry.

Fundamental Concepts of Euclidean Geometry

In uncovering the secrets of the fundamentals of geometry, Euclid realised that first must come the very basic fundamental concepts. Certain questions had to be answered to define these fundamental constituents of the field, such as what is a point, or what is a line?

Points

A point P(4,3) on a set of x-y axes, John Hannah - StudySmarter Originals

Lines

Euclid also defined something called a line. This he defined as "a length without a breadth." In essence, just a 1-dimensional segment with a finite length. He posited that a line could be extended infinitely in either direction. This is an area where modern geometry differs from Euclid's fundamentals, as now we refer to an infinitely extended line simply as a line and Euclid's line of finite length as a line segment. This is an important distinction to remember for the sake of correctness. Let's take a look at the difference below. Euclid also defined something called a ray, which similar to a line is infinitely long, however, it has a defined starting point. It is also sometimes known as a half-line.

A line on a set of x-y axis, John Hannah - StudySmarter Originals

A plane, in many ways, can be considered as similar to a line, but in two dimensions. A plane is simply a surface that extends indefinitely. A plane can exist in 2-dimensional spaces as well as 3-dimensional spaces and higher.

Angles

Angles were defined by Euclid as "the inclination of two straight lines." This essentially can be described in simpler terms as the rotational distance between two lines or line segments that share a point, i.e how much would we have to rotate one line before it lines up with the other. The shared point is known as the vertex of the angle.

Two line segments forming an acute angle, A, John Hannah - StudySmarter Originals

Dimensions

Dimensions are an important aspect of the fundamentals of geometry, which specifically deals with spatial dimensions. Spatial dimensions in mathematics and physics can be defined as the minimum number of coordinates required to describe a point in that space. For instance, a line has 1-dimensions as only a single number is required to specify a point on that line. Equally, if you wanted to specify a point on an x-y axis you would need two coordinates an x and y coordinate and on a set of 3-dimensional axes, you would need a third coordinate - the z coordinate.

A point in 2-dimensional space, John Hannah - StudySmarter Originals

A point in 3-dimensional space, John Hannah, StudySmarter Originals

Area

Area is a measurement that describes the size of a certain 2-dimensional region. There are various formulas that can be used in calculating the area of certain shapes. A good way to visualise area is to divide up a 2-dimensional space into squares. The area of the shape is simply equal to the number of squares contained inside it.

Depiction of the area of a triangle on a set of x-y axes, John Hannah - StudySmarter Originals

Volume

Much like area, volume is a measure that quantifies the size of a certain region of space. Volume, however, quantifies the size of a region in 3-dimensional space. All 3-dimensional shapes have volume, and similar to area there are many useful formulas for calculating the volumes of various shapes. We can visualise volume in much the same way as area, but rather than using small squares, we count the number of small cubes inside a shape. The image below depicts a cube in 3-dimensional space. How much space does the cube take up? Well, by counting we can see that the cube takes up the space of 64 smaller cubes, each with a volume of 1 units³.

Units

An important part of the fundamentals of geometry is the use of various units. In geometry, we use two basic types of units: units of length and units of angles.

Units of Length

There are two primary systems for units of length. These systems are the metric and imperial systems. The metric system deals in units of centimetres, metres, kilometres etc. whereas the imperial system works in units inches, feet, yards, miles etc.

Length is a 1-dimensional unit, however, units of 2-dimensions (area) and 3-dimensions (volume) exist which are composed of these units of length. The convention for the naming of these dimensions is shown in the table below.

Units of Angles

Degrees and radians are the two main units for the measurement of angles and it's very easy to run into problems if the distinction between the two isn't clear!

Firstly, it is important to recognise that degrees are an arbitrary unit of measurement that simply arose from the fact of the Earth's rotation. Ancient people watching the constellations in the sky move on a yearly cycle figured that since there were 360 days in a year (there are really 365 but they got pretty close!) that there should be 360 degrees in a full rotation. This has proved a simple and intuitive way to discuss angles as human beings, after all, we aren't computers and sometimes closely packed decimal numbers can be confusing.

However, since these early scholars sat looking at the stars, we have discovered another, arguably more mathematically sound way of describing angles. This unit is known as the radian.

Radians, rather than being related simply to 'amount of rotation', are related to distance travelled around an arc. Really, radians are in fact the distance travelled around an arc divided by the distance to the pivot point of that arc. If we take the equation relating a circle's circumference to its radius, we can find how many radians in a full rotation of 360^o.

$C=2\mathrm{\pi r}$

$\frac{C}{r}=2\mathrm{\pi }$

So in 360^o there are $2\mathrm{\pi }$ radians. From this, we can see that

It is important when using a calculator if it is set to treat your angles as radians or degrees when dealing with trigonometric functions to obtain the correct answer. Technically all mathematical functions taking angles as inputs work in radians.

The Fundamental Principles of Euclidean Geometry

Euclid made five fundamental postulates when delving into the field of geometry. These postulates were fundamental principles of geometry that he held to be self-evident, and informed all further principles and concepts of geometry thereafter.